Welcome to Think Tech on Spectrum OC16, Hawaii's weekly newscast on things in matter-of-tech and Hawaii. I'm Jay Fidel. And I'm Helen Dora Hayden. In our show this time, we'll review the top five Think Tech talk shows and the staff pick from a week ago. We'll check out the elements of the best of the best and get a handle on the coverage involved. Think Tech produces some 30 talk shows every week in its downtown studio. Here's what the studio looks like. And here's a list of all our incredible Think Tech hosts. Every week, Think Tech chooses its top five Think Tech talk shows from the week before based on the number of views each of them had on the internet. For this past week, the winning shows were swallows. Number one from the series called Bigotry in America, in a show entitled The Incident in Davis, California. With host Jay Fidel, that's me, and guest Gail Rubin, a retired attorney in California, and Shereem Kedosi, a Muslim journalist in California. Amir Shaheen from the Islamic Center of Davis gave a one-hour sermon on calling for the apocalypse against the Jews. And this is not just in Davis. This is elsewhere. This has happened at least two other times around the same time frame in California. He gave a one-hour speech. The mosque defended him. They put out a statement in defense of him. And after one week of pressure, including launching a petition, including getting the support of local elected officials like Brad Sherman, getting a bit of media coverage, the community finally came forward and he apologized for his statement. But when he apologized, he had the entire interfaith community used as props behind him. And the apology, which I use very generously in quotes, was pretty much a statement about how he's been a model for the community. But his one-hour speech is really indicative of where he stands in the interfaith community and where he stands as a Muslim and where he stands as an American Muslim right now in the landscape, which is very charged politically and on racial lines and religious lines. This is unacceptable in my eyes. Yes. Well, what did he say? What are the words? What are the thoughts that he expressed in that speech? The general language that these individuals have in the let-gale attitude to have covered this with other imams that have toured in Europe, that these imams travel from country to country, and we have them homegrown here, their language and their rhetoric is very apocalyptic. And I use that full knowledge of what that means. That means that there will never be peace for Muslims until Muslims fight with the Jews, that the last day we'll call on Muslims fighting with the Jews, and that until we get to that point, there will be no resolution, whether it's in Palestine, which is the issue that kicked off his sermons, whether it is Muslims living abroad and elsewhere in other parts of the world. And so it's very violent, it's very hateful, it's completely anti-Semitic, but I would go further than anti-Semitism. I would say it's full out, Jew Hatred. So he spoke two Fridays in a row, Friday, July 14th and again on the 21st. And he repeated essentially from the same hadith. And before I quote it, it's important to know that this hadith is also incorporated into the Hamas Charter. Hamas, as we know, is a designated terrorist organization that seeks the utter destruction and annihilation of the Jewish state. So these are the same words. And I quote, he quoted, O Allah, he's invoking the name of God to liberate us from the filth of the Jews. O Allah, show us the wonders of your ability that you inflict upon them. Show us the black day and count them one by one and destroy them down to the very last one. Do not spare any of them, do destroy them and do not spare their young or their elderly. O Allah, turn Jerusalem and Palestine into a graveyard for the Jews. Now I think all people of good faith and good will would agree that this kind of language does not belong in any civilized society and most of all not in a house of worship. And the fact that he has misappropriated a house of worship to utter these despicable, murderous words, we should all be outraged. And I am a Jew, an American Jew, and I live a mile away from this mosque. And this mosque is also located directly across the street from the University of California at Davis. And a large percentage of students that you see Davis attend this mosque. That should be of concern to everybody, why? Because now, if students are inclined to become radicalized, of course we know the internet is a large source. All they have to do is walk across the street and listen to Imam Shaheen. Number two from the series called Habachi Talk, in a show entitled Going to My Chiropractor with Gordon Bruce and guest Dr. Tanaki Berman. My specialty is all of physical medicine and exercise and how they interact to, not everybody does both with me, but I always have the PT and the personal training exercise tools to help people, that's what they really need to stabilize their adjustments. So very, very synergistic, you know, basically if you get the person all aligned, well don't you want to back that up with some specific exercise to stabilize you there so that, you know, I don't want to just keep fixing the same thing, you know, I want to teach you how to prevent things in the future. Balance. Yes, and you know performance enhancement for athletes, there's a lot of ways it goes, but you know, you can't tell people what to do, what I prefer to, you know, have it be synergistic where the patient is helping themselves and they're doing their exercises as well. You know, you don't have to, but like I said, it's a very good combination. It's the kind of combination of the three. So let's talk about one of the questions that I have is like, and I have two, but the first one I'm going to ask is, how has technology changed your patients? I mean, you know, we're a lot different now when we're coming to see you. So because you have technology to help you with the insistence and treatment, but how have we changed how you react to us? Okay, well, what, you know, this is something I've been thinking about how, because I get frustrated, it used to be pretty simple. You show a patient, this is how you need to set up your workstation. You know, in the past, I've gone into companies and done lectures on this, you know, to avoid work comp, you know, and some of it's so simple. But now it's like, well, why is your monitor, if you're sitting here, why is your monitor over there? Well, because that's where the outlet is. That's where the cables are. So in other words, all the workstations are getting set up to accommodate the technology rather than the actual person who's working. And one of the, I mean, there's a lot of problems with this. Like I said, a lot of people are working at right angles to their monitor, which is insane, absolutely insane. But the other phenomenon that happens, and this is a recent study on back pain, you know, like what can we do to prevent back pain in the workplace? Where is it coming from? Okay, well, the obvious is people that are repetitively bending over and lifting, right? You know, physical jobs where you lift a lot of things in front of you. But equally now, the problem is what's called creep. You sit, you don't get up, and the ligaments supporting your spine literally stretch out like old elastic. And you lose. Number three from the series called Perspectives on Global Justice. In a show entitled Don't Panic, Be Prepared with host Beatrice Cantelmo and guests Robin Lewis and Gordon Bruce. With the 2011 earthquake and tsunami and the Fukushima nuclear crisis, we saw that, you know, no system is lawless. You know, there are always areas of weakness.  And so what we saw was, you know, the government buildings and several government services were actually affected themselves. So, you know, if you go to the coastal areas, which were hit hardest by the tsunami, several of the government agencies and the local authorities were unable to respond or severely hampered in their response. And so, you know, in many ways, especially right in the beginning, the emphasis in many ways was on the community and the people themselves. So, you know, I think one of the big lessons is, you know, yes, the government is there and yes, they are well, very well prepared. But that doesn't always mean that you can rely 100% on the government. You know, we as citizens have to be ready and have to have a plan for ourselves as well. That's one of the key lessons. And I think another lesson was that, you know, it's too late to have conversations after the fact, after earthquakes happen. We need to be having conversations and building strategic partnerships, you know, between government, between the private sector, between NGOs. And well before these things happen, that when they happen, we are in a good position to respond. We have a household level Desire to Preparedness Training Program, which essentially targets families. So we found that, you know, if you're at work, there's usually some kind of plan at the workplace. If you're at school, the school should have an emergency response plan.  So we thought, you know, why don't we target family and help them make their own disaster plan so that when something happens, they know exactly where to go. Number four from the series called On the Street, in a show entitled Hawaii's Feral Cat Problem, is killing cats the solution with host Carol Cox and a number of guests on the street. I'm down here at 4th Street Mall, Think Tech, Hawaii. And we're just asking people along the way about these proposed rules that the governor is considering allowing the Doe Bar, the Department of Boating and Recreation, DLNR to actually shoot cats, kill them in any manner that they desire. It's pretty horrible. Yeah? Yeah, that seems extremely inhumane and immoral. Are you a cat owner or a pet owner? Pet owner, not cat. I have a dog. Yeah, this is the concern that they want to sign this bill, want the governor to sign this bill and be able to just shoot, poison, trap, euthanize anything. Oh, that just sounds completely awful. Wow, I can't believe that they would propose such a thing.  And then if no one wants them, then do what we've been doing for years by putting them to sleep if nobody wants to adopt them or take care of them. So you would vote on the side of... It's inhumane in a certain way. It's like if you don't want us shooting people on the streets, why are you going to go and shoot a living thing? It's also, I mean, I guess we hunt also like deer or whatever, but I mean, it's kind of wrong. Well, I think they should issue permits to the hunters. Yeah, and not do the random killing. Not just randomly kill them. Yeah, yeah. Okay. And not let the animals rot, I think that's part of the controversy. No, I don't think they should just be able to go free range hunting cats. I think there's a better alternative. What would you suggest to them to do? I don't know what they do now. But I don't know if there's any kind of group or coalition that goes around and, you know, cages them or someone who would, you know, like the human side of you, you can call them. I don't agree with that. I have three cats myself and they were all, they're all feral. I got them fixed through Popoki and, you know, there's ways to get them fixed. Where you don't have to spend, yeah, poydogs in Popoki and they fixed my cats and they're happy cats now. So I mean, they're still living, they're still living, you know, animals that we have to take care of. And I don't agree in shooting them to get rid of them, but I think we have to take care of the problem of them multiplying by taking them to those nonprofit organizations like Poydog and Popoki and everybody do their part to take care of the matter that way. Well, I think there needs to be humanity with us. So they need to be, I'm not in favor of just shooting animals. Number five from the series called Where the Drone Leads in a show entitled The Spreading Tapestry of Dronism, with Ted Rawlston and guest John Franklin. So the matchmaking function within the electronics is okay on high density. It's not going to bother it. I don't think so. I mean, to be honest, in arm testing, we tend to be pilot limited and I haven't been able to put up 15 at a time.  We don't have enough drone pilots to serve the commercial industry if they ask for one more person. And we're using airline pilots and this sort of thing and reprocessing them in this domain and together we have to come up with an educational framework that will produce educated people and train people, certified people in the world of flying and the world of design in the world of counter drone. So it's a really exciting time getting your 107 and actually flying a mission successful mission. Yeah. 107 has its own downsides that are subject to yet another discussion called liability. But in any case, talk about the company. What else you make besides the electronic detection systems? So we started life as a passive acoustic detection company. So we've always been a counter drone business but about two, three years ago we were using microphones to do it and microphones are actually an interesting way to do it. I can't tell you how many times, like for example in my story when I was on vacation, the first thing I do is I hear it. I don't see it and maybe it's just I've been doing this so long that I've tuned into that but without a doubt I'm going to hear it before I see it. We also have a staff pick. This time it was from the series called Law Across the Sea in a show entitled All the Worlds of State and All the Men and Women Merely Lawyers with host Mark Shlove and his guest, Attorney Alan Ma. Well, being an immigrant and I and of course I also speak three different dialects of Chinese and I was very involved in the Chinese community and I saw a lot of people they really needed help and they couldn't command English and they didn't understand American law and often because of their ignorance and they put themselves in difficult positions. So that kind of motivated me to become a lawyer, to be able to help some of those people in the immigrant community. And you saw these problems as you were growing up here in Hawaii after you arrived. That's correct. Okay. And that kind of inspired you, is what I'm hearing, to look towards the law, look to a way to help people. That's correct. So, okay. So what happened next in your life? You went to law school and you were here in Hawaii but you went away to law school. Right. Well, I was very fortunate. I was awarded this national scholarship called CLEO, essentially Congress and at least in those days they select about 250 students nationwide to receive scholarship to have a special assistant for them to go to law school. But those scholarship recipients must commit to going back to their own state. In my case, I came back to Hawaii in part because I received a scholarship and my commitment was to help the people in Hawaii. And so after I concluded my legal education, I came back to Hawaii to work. The scholarship sounds like a pretty good deal, pretty good idea. Is it still going on? I believe it is. And I believe the program is still going on. Okay. So, all right. You got the scholarship, you finished law school then. What happened in your career? Where did you, you had this feeling about the immigration law? Where did that take you? Of course, when I started looking for jobs and I knocked on doors at that time and all the prominent immigration law firms or lawyers and I came across my then employer who became my mentor and subsequently my law partner, Mr. Ronald Odenberg. If you weren't able to write these links down, you can always find them in our daily email advisory. If you don't get our daily email advisory, you can sign up to get them on thinktechhawaii.com. These are only samplings from the top five in the staff pick from our 30 weekly talk shows. There are of course many more. To see these shows in their entirety, go to thinktechhawaii.com or youtube.com slash thinktechhawaii. Great diversity, great community coverage and great content at ThinkTech. Yep, every day better. And now let's check out our ThinkTech schedule of events going forward. ThinkTech broadcasts its talk shows live on the internet from 11 a.m. to 5 p.m. on weekdays. 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More than ever, ThinkTech lives on the internet and yes, in its ongoing efforts to push the innovation envelope, ThinkTech is getting into crowdfunding and also into animation. I'm Rachel with ThinkTech Hawaii. Help ThinkTech Hawaii. Why? Because ThinkTech helps you and the people of Hawaii by giving a free platform to share information, ideas and news on matters important to our community. We want to make Hawaii a better place and raise awareness through citizen journalists like our volunteer hosts and public engagement. How do we do this? We produce and stream 30 video talk shows a week on our site thinktechawaii.com and post them to YouTube, community TV and cable and as podcasts. We do this from our downtown studio and from equipment that live streams from anywhere in the state. We work to find the best of Hawaii and let everyone know about it. To continue to do this, we need to raise money to keep our studio and studio staff going and to go to the neighbor islands and cover people and stories there. So this November is Give Thanks for ThinkTech Month. You'll be seeing more about this campaign and more about our efforts to animate. Why don't you tune in to thinktechawaii.com where community matters. They're rolling out a new show. It's called Trump Loy. Guess who that's about? Check it out. Go ahead. Give us a thumbs up on YouTube or send us a tweet at ThinkTechHI. We'd like to know how you feel about the issues and events that affect our lives in these islands. We want to stay in touch with you and we'd like you to stay in touch with us. Let's think together. And now here's this week's ThinkTech commentary. Okay. We all have problems. We all have so much to do. How are we going to get it all done? How are we going to cope? There's a very famous saying by Confucius, the man who moves a mountain begins by carrying away small stones. I often wonder if I have what it takes to finish what I had started. I'm sure you have heard the expression making mountains out of molehills. Well, let's see if we can make sense of moving mountains. Whether it's your body or your mind that is forced to ask the question, can I do it? Let's try to move that mountain. How did we get here? The road to this mountain of reflection is long and arduous. All we are doing is living our life and face the pressures of doing the best we can, right? In life, we all tackle mountains that test both our mental and physical strength. To do that, and to do it well, we have to move a lot of small stones, endure pain, jump over barriers, break ceilings, work hard for hours, and deal with discouragement. The key is to clear your head and process your situation. We have to weigh the pros and cons and dig deep for some answers. You have to come to a place where you know what to do. It really becomes an epiphany moment. Quitting it is not an option, right? Remember one small stone at a time. You have to be present where you are, not get ahead of yourself, and start the journey. We'll be right back to wrap up this week's edition of Think Tech. But first, we want to thank our underwriters. Grateful thanks to our underwriters. The Annie Sinclair-Nutzen Memorial Fund. The Atherton Family Foundation. The Bernice and Conrad von Hamfun. Kasselin Cook Hawaii. The Center for Microbial Oceanography Research and Education. Collateral Analytics. The Cook Foundation. The Hawaii Community Foundation. The Hawaii Council of Associations of Apartment Owners. Hawaii Energy. The Hawaii Energy Policy Forum. The Hawaii Institute of Geophysics and Planetology. Hawaiian Electric Companies. Gailin Ho of BAE Systems. Integrated Security Technologies. Kamehameha Schools. Carol Mon Lee and the Friends of Think Tech. MW Group. The Omediar Ohana Fund. The Schuyler Family Foundation. The Sydney Stern Memorial Trust. Yuriko J. Sugimura. Thanks also to our viewers like you. OK, Helen. That wraps up this week's edition of Think Tech. Remember, you can watch Think Tech on Spectrum OC16 several times every week. Can't get enough of it, just like Helen does. For additional times, check out oc16.tv. For lots more Think Tech videos and for underwriting and sponsorship opportunities on Think Tech, visit thinktechhawaii.com. Be a guest or a host, a producer or an intern and help us reach and have an impact on Hawaii. Thanks for being part of our Think Tech family and for supporting our open discussion of tech, energy, diversification and global awareness in Hawaii. And of course, in the ongoing search for innovation, everywhere you can find it. You can watch this show throughout the week and tune in next Sunday evening for our next important weekly episode. I'm Jay Fiedel. And I'm Helen Dora Haydn. Aloha, everyone.


Hopefully I will be able to join my colleagues again, at least on Zoom. I have, let me tell you a little bit about my background and I came to the legislature having worked for 25 years in community mental health. I'm trained at a master's level in counseling psychology. I worked as a substance use disorder counselor and founded a foundation, film tropic foundation, the Samara Foundation of Vermont which is now part of the Vermont Community Foundation as a major fund. And I brought that background into the stay house because I felt like I wanted to make bigger changes than could be made as, even though profound changes happen when I sat individually with clients and then when I became the executive director of the counseling service of Addison County, I found the metaphor that I always found myself thinking about was I want to paint on a larger canvas because as I met with individuals, I knew policies that made it hard for them. I then was the middle manager and then the executive director and even as executive director, there were policies that needed to change at the state level. And so I organized statewide groups for substance abuse counselors. And they then said, someone needs to go to the stay house and make that change.  Well, that was what got me to the stay house for the first time. And I have had the good fortune to serve there for nearly 25, 26 years. I've been able to be part of making some major changes in social policies. And each time that I find, and not alone, of course, but with my colleagues and each time that I've found that we've touched people's lives by actually making change at a policy level at the state, it just renews and rekindles my desire to be part of that. And so that's why I'm wanting to go. I've been, again, I've been appointed by three different speakers of the house as the chair of first the House Judiciary Committee and now the Health Care Committee. And being elected as a trustee of the Vermont State Colleges Systems twice by my colleagues. So I feel like I bring both a depth of experience in my personal life. And I brought that to the state house where I have learned to build relationships that help move social policy forward. That's a broad general description of why I'm interested, but that really is my motivation. I want to be part of making change. Well, I'm glad we have the time to go into depth on this because it's an interesting story to tell. And I wonder when you look ahead to the next session and the unique challenges of this time and also the challenges of our state, what's your priority? What's important for you to continue working on? Well, several things come to mind and nothing could be higher on the agenda right now than supporting Vermont and Vermonters around the pandemic. This COVID pandemic has upended so many people's lives. It's actually upended all of our lives. And I think partly because of, perhaps because of some of my background in mental health, I'm also acutely aware of the stress that each of us is operating under. Even those of us who are fortunate to have enough food, to have enough shelter, to have transportation if we can ever use it. But there are many of our fellow Vermonters who are reaching the end of their unemployment benefits who are struggling with shelter, who are struggling for without sufficient food or money for food. And so at the personal level, that has to be top of our agenda, making sure that Vermonters are safe, healthy and secure in terms of COVID-19. We need to ensure that the healthcare system of Vermont continues to be able to respond nimbly to the changes in the pandemic. And at some point, I'd love to talk a little bit more about what we've done, which I think has been extraordinary actually. And under the leadership of our governor, who I give credit, we are safer than almost any other state in the country. So COVID-19 response to that. The second thing I look forward to really working on is continuing our long-term commitment to social justice, particularly around racial inequalities. One of the things that became so horribly clear in the midst of this pandemic is how the social inequities in our society, particularly around race, played out both in Vermont, but even tragically more so elsewhere, but absolutely in Vermont, where people of color who contract, who are much more vulnerable to serious illness from COVID-19. And it just really reveals the underlying social inequities in our healthcare system and in our social system generally. So I want to continue to be part of initiatives that are trying to address systemic racism, eliminating systemic racism, unbuilding it, taking it apart. And that starts by acknowledging it and we won't change it in one day or one with one bill, but that's another one of my priorities. You're muted, so. Thank you, everyone is very right on the mute. That's great. We encounter it regularly in our legislative life, after we mind each other all the time, you're on mute. That's funny. My question has to do with the substance abuse epidemic, which I would even say is beyond opioids. I think people are under enormous stress and are having a real struggle with addiction. And I wonder what's happening in the legislature, what will you be working on your committee to address this? Interestingly enough, this is an area where the legislature, I think is not structured in the house in the way that it might be. Substance abuse or substance use disorder issues actually are under the jurisdiction of my good colleague, Representative E.M. Pugh, who I believe will be part of this sometime soon. While I chair the healthcare committee and we do now have jurisdiction over the issues of mental health, jurisdiction for substance use disorders are in a different committee. Nevertheless, I have worked in that area, both as chair of judiciary and in conjunction with my colleagues. I'm not surprised, but I'm really dismayed. When I hear how much additional alcohol is being consumed during this period, I heard something that was like 25% more alcohol is being consumed. We already, opioids is a scourge and just in the last, literally the last day, I think we've heard about a settlement with the pharma company that promoted, wrongly, the highly addictive opiates. But I think I continue to hold out, even as we have to address the opioid epidemic, people need to recognize that alcohol is still the number one drug of abuse. When I was the chair of judiciary, I actually with legislative council would insist that when they drafted a bill for us and not say alcohol and drugs, that it say alcohol and other drugs. Because alcohol is a drug. It happens to be the drug of choice. It happens to be the socially acceptable drug for many, many people. And yet the damage that is done by alcohol within our society is enormous. Both in terms of personal lives on our highways, someone once told me that very, very high percentage of all the emergency room visits have some type of alcohol related involvement. And so I have to say I'm a little bit out of touch with specifics around what's happening with the opioid epidemic at the moment. But I've heard indirectly that in fact, things have gotten worse, not better. And that doesn't surprise me either. We were really focused as a state on the opioid epidemic. And we really did some, made some cutting edge changes that were actually mirrored by a lot of other states. The hub and spoke model, which really made opioid treatment more available to more of our monitors, particularly in a rural state, engaging primary care physicians, as well as hubs where medications could be prescribed that couldn't be prescribed as easily by physicians. So I think we need to continue to put some attention there. It's challenging, I'll just say this. We're a citizen legislator, legislature. We, I think we do a pretty amazing job at juggling many difficult issues at a time. But this is one which I think has started to slip off of our attention and we can't afford to do that because Vermonters continue to suffer. I have constituents who approach me about family members and how to get help. I think most, every one of us knows someone whose life has been disrupted by, if not opioids, certainly other drugs, including alcohol. And again, I don't want to sound like a prohibitionist, but I do think we need to recognize that as much as alcohol is a source of enjoyment and relaxation for many people, it's also the source of enormous social disruption. What's your view on the marijuana legalization or commodification in Vermont? Well, to be quite honest, I've not been a fan of promoting it, of promoting, I supported legalization because in fact, it was so clear that legalization was disproportionately impacting Vermonters, and I'm gonna talk about Vermont now for, they was disproportionately impacting Vermonters with consequences that didn't make sense as we came to understand that marijuana was being much more widely used and the federal government was actually had pushed for years to make it, to have high penalties. And then I think some of us began to understand that there was a way in which marijuana nationally, again in Vermont to a degree, but nationally as well was disproportionately impacting communities of color. And so I supported as the Chair of the Judiciary Committee, in fact, I was the Chair when we decriminalized small amounts of marijuana. I knew full well at the time that that was going to lead to a conflict because, okay, now it's not illegal, but how do you get it legally? My background as a substance use disorder counselor has led me to be very concerned about the way that, particularly the way that corporate America markets drugs. And again, I'm calling alcohol a drug, which it is. And what I've seen happen with alcohol is for me, there's just no shame, no level of shame that the marketers will have in promoting the use of alcohol. For me, the final straw, I mean, I don't know what the final straw was, it was like when they started to promote something called alcohol, which was powdered alcohol. They said, well, now you can take it when you go backpacking, now it's like, and the claim was, oh, youth won't use this and college students won't use this and no one will abuse it. And next, within weeks, within months of it coming onto the market, stories of people snorting it, people having difficulties from it in emergency rooms, and we banned that in Vermont. We banned the sale, but we were one of the states that stepped up and said, no, you cannot sell that in Vermont. And I'm concerned, so I was reluctant, I was not one of the people pushing to have a legal regulated market, but I came to realize that it's here. People are using it, people are buying it, better to have a regulated market, particularly when fentanyl, when fentanyl laced marijuana became available, it was not a came available, it was around, we have deaths, you don't, what are you buying? Now, of course, we're in conflict with the federal government still, but I, we took testimony, I said, okay, if that bill is gonna move through the legislature, it's gonna have to be, have testimony in the healthcare committee as well. And we took testimony and added provisions in our committee, which were an attempt to protect romances from high dose, high unregulated dosages from, we actually supported having the health department make the decisions about advertising and a number of other provisions to try to, I finally decided strategically, one of the things I learned over the years, if you wanna affect something and you're not a big fan, decide to support it and try to make the change. You're more likely to make successful changes if you're supporting it, and I knew it would pass. There was not a question that it would ultimately pass at some point, and so I tried to improve a bill that I wasn't thrilled about, but I did vote for, I say I voted for it, because it included a number of provisions, which I thought we're going to be helpful in keeping it out of the hands of youth. My big concern is that when we get, we talk about a big pharma, big tobacco, I think it won't be long before we're talking about big marijuana, and I, the small growers in Vermont say, well, we need to be part of the market, and the committees tried to set it up so that would help. I'm not fully persuaded that we've been successful in that effort, and I think there's a lot of money to be made here and where there's a lot of money to be made, I think we need to be very vigilant and I intend to be. Lisa's for what part I can play. Thank you, that's an interesting insight into how politics works also, not only. Yeah, well it is, it's because it's like I, and I had a constituent say to me, in a rather confrontational manner, I don't know, I can believe you anymore. You said you weren't going to support legalism, you weren't going to support tax and regulate marijuana, and now I see you doing it. Well, I tried to explain, and I tried to explain in some written materials to my constituents. So they were disappointed, but I made that decision and it's one I'll stand by. We have time for one more question. So do you want to talk about the COVID accomplishments or is there something else you'd like to say? It is time flies by, doesn't it? I know, no, we can talk for an hour. Yeah, and I would love that. Because actually I found myself thinking, I would love to think of this as a conversation because there's a lot to talk about. Well, let me say a couple of things. Let me just look here and just make sure that I'm, well, first of all, Vermont received an extraordinary amount of federal assistance, $1.25 billion. We don't usually use those numbers and throw them around as easily. Just yesterday, I'm also, I didn't mention that I'm a member of the Joint Fiscal Committee. We met this morning and the report is that, well, one of the things that we did between our committee and the Human Services Committee that represented Pew, we worked collaboratively and we crafted with the administration collaboratively a proposal to have, I think it was about $250 million go into helping to support the healthcare infrastructure in the state of Vermont, which stood itself up in record time. If you remember, we were setting up surge hospital beds. We were all elective, all elective procedures were dropped. So that, and that is where unfortunately in our current fee for service setup, the hospitals were bleeding dollars, medical practices. I've talked to someone in my community who works with a private, you know, primary care physician practice.  Now the good news is that didn't materialize in the same way, and that we are so incredibly, it's not just luck, it was good leadership and good implementation by Vermonters who took this seriously, but the result was we lost hundreds of millions of dollars in our healthcare sector, which we cannot afford to do while we're in the midst of a pandemic. And so I want to give credit to Speaker Missy Johnson who on the middle of March, I think it was actually it must have been March 12th because it was Friday the 13th, when we on March 12th she called a special meeting of all the chairs of all the committees, including myself as the chair of healthcare. And she said, I want each of your committees to take a look now what kind of emergency COVID legislation we need in the next 48 hours. And I want you to have legislation that you can pass out of your committee by Friday evening. That was like two days. I mean, that was an extraordinary challenge. And I have to say that the partners that we worked with, we worked with a group of nonprofit healthcare leaders who stepped up and we passed within 48 hours an extraordinary piece of legislation which authorized the secretary of human services to suspend all kinds of rules around, you know, like nurses can come into Vermont if they usually it takes a while to get in here. We authorized them to move money out to nonprofits which we're going to go under because we appropriated, let's say, you know, emergency, I think it was $150 million that the administration could appropriate themselves on an emergency basis. And then we said the joint fiscal committee would be involved for further things. And then we, when we came back, we appropriated the $250 million for healthcare. And $87 million has gone out the door as of yesterday. There's a round two and we made sure that small healthcare providers could get access. And so we deliberately said it's not going to be first come first served because we knew the larger organizations which need to have a larger infrastructure because of the complexity of what they do could get at the front of the line quickly. And we said, no, we need small primary practices. We need dental practices. We need an acupuncturist who's in a private practice. We need other healthcare providers to be able to submit what their losses are because of COVID, have the agency of human services sorted through and as of yesterday, they have appropriated or they have spent and sent checks out for $87 million. That still leaves us over $100 million. And there's a round two that's just starting because we know that need will now, now they can apply for money through, I think September 15th and there may be around three. I think, you know, we can be proud of how Vermonters responded to this pandemic at many levels. And most Vermonters will have any reason to pay attention to what I'm describing. But I am very proud of the Vermont state employees who really, they brought in employees from all parts of state government. This is a major undertaking. We engaged a federal or a national group to help us monitor to make sure we weren't spending money where we shouldn't be spending money. And there's a lot more that could be said about this but I'm very proud of that. In addition to which I'll say one other thing, our committee in healthcare had started to work on promoting or making telemedicine more available. We did that before there was a COVID and we had a bill already in the Senate from the house. And so when COVID suddenly hit like it did and we knew telemedicine had to be part of this, I felt very, very proud that our committee had already laid the groundwork and we folded that into the emergency legislation and actually promoted it even further. I think there's a lot of stories to be told about housing, how the committee that deals with housing responded to the pandemic. And I think we're going upstream and making changes so that the homeless population of Vermont, both on an emergency basis was taken care of and hopefully we'll do some structural changes. That was one of the other directives from the speaker. Think about how to use the COVID dollars, not just to jam them out the door, but think about how we can do something that will structurally change. This is an opportunity that will not return in our lifetime. Hopefully, now hopefully we'll get another stimulus package if the Congress can get it together. But we are still working on how to responsibly use the 1.25 billion dollars. Just yesterday, I'm a trustee that run state colleges and I'll just, and just yesterday I learned that 2.3 million dollars was appropriated for emergency free courses with free support and computers if necessary. And I learned yesterday from the head of the state colleges that that 2.3 million dollars is gonna be completely used over a thousand Vermonters applied in short notice for 3,000 different special free offerings to help them with get their employment back and again, it's another way to support the Vermont state college system and the state college system supports Vermonters. I think there's a lot to be said. Usually you're muted again. We're touching, yeah, we're just touching the surface of the work that happens in the legislature and that you have been doing in your capacity and leadership and also as a representative from District 4-2 for Heinsberg. So I wish we could talk all day, but maybe we could do some talking during the legislature and get an update. I'd be happy to do that. Can I say one last thing? I don't know if you have a minute more. Go ahead. Just yesterday, this is, I'm gonna take, just yesterday, Pope Francis announced that he thinks civil unions for same-sex couples are appropriate. A huge change. 20 years ago, I as the vice chair and Tom Little as the chair sat in the House Judiciary Committee in Vermont and coined the term civil unions. There's some, knowing that we finally achieved full marriage equality, but knowing that we helped set something in motion that 20 years later would affect one of the largest influential institutions in the world gives me great satisfaction. And I think Vermonters should take great pride in many of the work, many of the things that we've done that we're very forward-looking. And I think we're at a place to do some of the same now. I'd love to come back and talk more. Bill Lippert, thank you so much. Representative Bill Lippert from District 4-2, representing Heinsberg, he has an opponent, which is Sarah Toscano, Rony Toscano. Sarah Toscano, running as a Republican. Don't forget to vote on 11-3, November 3rd. Yes, don't forget to vote. If you haven't already, don't forget mail-in ballots are due on the 24th, but it's Saturday. So depending on when you're watching this, if you have not already mailed your ballot, bring it down to the voting place or to your town clerk. Now or before November 3rd. So thank you for your time, Bill, and your service. And thank you very much for watching. Thank you.


Look, I think that the Fed has a lot of latitude here. They can focus on low inflation and say they want to do nothing, or they can focus on the idea that, listen, it has nothing to do with inflation. It has to do with the fact that the economy is strong enough that we've got to get back to normalized rates. And that's what they're going to do, which is why I think the banks are still a good place to be. We're making some moves and action alerts on the banks. You've got to stay tuned. But a lot of us have to do with the fact that they've had a monster run anticipating the fact that the Fed minutes, I think you're going to show that they've got great latitude. You can't keep going up on the same news as the C-CAR and interest rates going higher. But remember, the Fed is in a great position here. And they waited, and congratulations to Janet Yell, it made a lot of sense.


Welcome to the GVSU Calculus Screencasts. In this episode, we talk about limits of sequences. Recall that a sequence is just a list of terms in some specified order, or, more formally, it's a function whose domain is a set of positive integers. In Calculus I, we describe what it meant for a continuous function to have a limited infinity. And now we're going to see how to translate that idea of a limited infinity for a continuous function to a limit of a sequence. So let's go back to Calc 1 and review the idea of a limit of a function, a continuous function at infinity. Let's consider the example of the function f, defined by f of x equals 1 over x. And the graph of 1 over x is that familiar hyperbolic shape. Now pause the video for a moment and consider if f of x has a limit at infinity. If yes, why, and what's the limit? And if no, why not? Resume the video when you're ready. Remember that if we can make the values of f of x as close to some number as we want by choosing x to be large enough, then our function f has a limit at infinity. In this case, we can make all the values 1 over x as close to 0 as we like by choosing x to be sufficiently large. So we say that the limit of f of x as x approaches infinity is 0. Now let's translate this into the sequence world, and let's consider the sequence s sub n, where s sub n is 1 over n for each positive integer n. We can plot the sequence as a function whose points are of the form n s sub n. After all, a sequence is a function from the positive integers to the reals. For example, the point 1 s sub 1, which is a point 1 1, is plotted here, along with the graph of the continuous function 1 over x for comparison purposes. We can plot more points in the sequence to get a sense of the behavior of the sequence, and we'll plot the first five points in the sequence on the next few slides. Here's the first point, 1 1. This is the second point, 2 and a half. Now 3 and a third. And 4 and 1 quarter. And we had 5 and 1 fifth. And you can probably see the process. Here we'll plot the first 25 points in the sequence. And here you can see that the sequence follows the same overall pattern as the function 1 over x, except only hitting the integer inputs. And notice that just as with 1 over x, we can make all the values in our sequence 1 over n as close to 0 as we like, as long as we go far enough out in the sequence. And going far enough out in the sequence means choosing n to be as large as we need. So we can translate the idea of a limit of a continuous function to that of a sequence by saying that our sequence, in this case 1 over n, has a limit of 0 because we can make the values of 1 over n as close to 0 as we like by choosing n to be sufficiently large. We can do this for any sequence. We'll say that a sequence s sub n has a limit of some number l. If we can make all the values of s sub n as close to l as we like by choosing n to be large enough. When this happens, we write limit of s sub n as n goes to infinity is equal to l. And we also say that the sequence s sub n converges to l. So as a concluding example, let's consider the sequence n over n plus 1. Pause the video for a moment and think about this sequence and make a conjecture as to whether this sequence has a limit or not. If so, why and what's the limit? And if not, why not? Here's a plot of points in the sequence n over n plus 1. And this graph seems to indicate that we can make all the values of n over n plus 1 as close to 1 as we like as long as we choose n to be big enough. And so that indicates that the limit of the sequence n over 1 plus n is 1. The main idea to come away with from this screencast is that a sequence s sub n has a limit l if we can make the values of s sub n as close to l as we like by choosing n to be sufficiently large. That concludes our screencast on limits of sequences and we hope to see you again soon.


In this lecture we will learn the following things. We'll take a look at mechanical and electromagnetic wave equations to inspire our thinking about matter waves. We'll learn how to infer the nature of the wave equation for matter from an exercise involving the conservation of energy. We'll look at the meaning of the waves described by the matter wave equation, the so-called Schrodinger wave equation. And finally we'll look at the limits of absolute knowledge that are imposed by the wave nature of matter. Let's take a peek at waves beginning with classical mechanics. An introductory physics class would have taught you about oscillatory phenomena. And a wave is just another kind of oscillatory phenomenon that can be described by time and space-dependent functions. So in introductory physics we learn that a time varying oscillation along one dimension, for instance a mass on the end of a spring that's bouncing back and forth on a friction the surface or up and down in a gravitational field, can be described as simple harmonic and nature. And this allows us to write a mathematical function involving, for instance, the cosine of frequency and time and an offset from the amplitude being maximal at zero. This is a typical equation you might see in introductory physics to describe an oscillatory phenomenon. Now here omega is a special kind of frequency. It's known as the angular frequency and it's given in terms of the period of oscillation, which is a more familiar concept. The period of oscillation, often denoted by capital T, is simply the time required for one cycle of the phenomenon to conclude. The angular frequency is 2 pi divided by the period. And this essentially means that it's 2 pi times the frequency of oscillation of the phenomenon. Angular frequency is the rate of angular displacement if we were to model the repetitive behavior as going around a circle, completing one cycle of the circle, 2 pi radians, as completing one cycle of the phenomenon. Now for all considerations here, let's set the phase angle, the degree by which we would need to offset the cosine function to get the amplitude to match the initial conditions of our oscillator. Let's set that phase angle to zero. Let's set phi to zero to simplify this equation.  And we pluck the string, that is, we displace part of the string vertically, then let it go and it bounces up and down and up and down, the vibration of a string. Now we have a distortion in y that's traveling along x in time. And the solution to that problem looks something like this, that the displacement in y at any position x and at time t is given by some initial y times the cosine of a spatial part, k times x, I'll come back to k in a moment, minus a temporal part, omega t, which we're already familiar with from the equation up here on a simple one-dimensional oscillatory phenomenon. Now what is k? Well, k in this context is known as the wave number. And it's defined by 2 pi, the number of radians in a circle, divided by the wavelength of the phenomenon. So you can think of this as describing the number of cycles per unit distance in the phenomenon, whereas the angular frequency is the number of cycles per unit time. But these functions answer some question. And if they're the answers to a question, what is the question? Well, they are all solutions to a wave equation. That is, an equation that describes how changes in space relate to changes in time. Now the one-dimensional mechanical wave equation, at least the one that tells you about vertical displacements and how they vary as a function of horizontal position in time, is simply given by the second derivative with respect to time of the vertical amplitude y. And that's equal to the constant squared times the second derivative with respect to space of the vertical displacement y. And y, of course, is a function of x and t. So if you try applying this wave equation to the solution on the previous slide, you'll see the following. First of all, the left-hand side is the second derivative with respect to time of the vertical displacement y. Plugging in our function for the vertical displacement, we would get this equation now. The second derivative with respect to time of our description of the vertical displacement versus x and time. Taking one of the derivatives of this function results in us having to do the derivative twice. First of the cosine function and then of the argument of the cosine function. Well, the derivative of cosine is going to be the negative sine function. And the derivative of the argument is going to return a negative omega, a negative angular frequency multiplier. And so we'll be left with this. Omega times the original amplitude y sub zero times a sine function of the original argument. We have to take the time derivative of this one more time. If we do that, we wind up with an additional factor of negative omega out in front of the original cosine function. And so at the end of this, we wind up with an equation that's just negative omega squared times the original function y of x and t. Now let's handle the right-hand side of the wave equation. This is a constant term squared times the second derivative with respect to x position of the displacement y. We plug in our function for y again. Now taking the first of the two spatial derivatives that we have here, we wind up with a function that looks like this. So the first derivative of cosine returns negative sine. And the derivative of the argument multiplied by that gives us a factor of k. And so we wind up with this. And we have to take the spatial derivative of this one more time. And at the end of this, we wind up with an equation that's negative the constant squared times the wave number squared times the original function y of x and t. Now setting these two things equal to each other, as would be required by the wave equation, we find out that the function y of x and t drops out of both sides of the equation, leaving us with this simple relationship between the angular frequency squared, the constant squared, and the wave number squared. And if we take the square root of all of this, we see that we have c equals omega over k. And this is one of the velocities that's present in mechanical waves. The speed of the mechanical wave is given by the ratio of the angular frequency and the wave number. Now this is a very quick tour of a solution to the wave equation and how you can see that it does solve the wave equation and how when you plug it in, it returns a relationship between frequency squared, speed squared, and wave number, which is related to wavelength squared. A dedicated waves course would spend a lot more time on this, motivating the derivation of the wave equation itself from a simple model of a vibrating string or something like that, motivating how one sets up and solves that equation, and then showing you what relationships emerge from solutions under different conditions. Here, I am merely trying to motivate some thought process about wave equations and the resulting relationships that can be derived from the application of those wave equations to their solutions. So sticking with mechanical waves for a moment, let's think a little bit about the energy that's contained in that wave. So again, our model here is a mechanical distortion of a physical medium. And that medium that I have in mind here might be a string made from many little bits of mass all hooked together as if by little springs, each with a spring constant and so forth. So if we model a string that way, we can think about the string as having a total mass, capital M and a total length, capital L. And the little bits of mass it's made from are all equal in size and uniformly distributed along the length L. And so this string has a uniform linear mass density given by the Greek letter mu, which is mass divided by length big M divided by big L. No matter what chunk of the string we look at, every chunk will have the same mu because it's a uniform distribution of mass. And so we can always relate mu to the mass in that chunk and the length of that chunk. If we then vibrate the string such that a given part of it at some time t in location x will have a small mass M and that mass will have a vertical velocity Vy, that velocity will oscillate transverse to its length just as the displacement oscillates transverse to its length. That tiny little chunk of the string will have a length dx, a differential of x, and a little mass M that can be related to the length dx by the linear mass density. So the little m divided by the little dx would be mu because it's a uniform distribution of mass. So that means that m is equal to mu dx. And every place we see m we can replace it with this product and vice versa. The kinetic energy of that little chunk in a moment of its motion as the string vibrates will be defined by its mass and its velocity at a given moment in time t. So taking the classical definition of kinetic energy, we're thinking about a mechanical wave here, so let's think classically for a second. We have the little bit of kinetic energy possessed of by that little bit of mass is going to be one-half times its mass, which is mu dx, times its transverse velocity squared, Vy squared. Well, Vy is just the derivative of the displacement in the y direction with respect to time, dy dt, and we're going to square that. So we wind up, if you plug in that derivative and do that as we did on the previous page, we wind up with this equation for the little bit of kinetic energy possessed of by that little bit of mass that makes up the string. So this is the term for the little bit of kinetic energy possessed of by that little chunk of mass. Now, because it's hooked to its neighbors by springy things, each mass is linked to the next by, you could imagine, a little spring with a spring constant. The potential energy stored in that same chunk of mass will depend on the elasticity of the string. That is the stiffness of the little springs that you could imagine hold one chunk of mass to the next. So thinking of the string this way is concocted of a whole bunch of little masses M connected to their neighbors by little springs with spring constants kappa. Then as an introductory mechanics for oscillatory phenomena, masses on a spring, you can relate the angular frequency squared to the ratio of the spring constant and the mass. That is to say, the spring constant is related to the mass times the angular frequency squared for an oscillating mass on the end of a spring with spring constant kappa. So the little bit of potential energy that's stored at that location at x in that little mass M is just one-half times the spring constant times the displacement from equilibrium squared. Well, that's just going to be a little chunk of potential energy held by that little bit of mass. One-half times kappa times the displacement squared. We go ahead and substitute for kappa with M omega squared and we can substitute for M with mu dx. And then finally we can put in our equation for the displacement and that now involves the cosine and that whole thing is squared. So we have kinetic energy. We have potential energy. Let's look at the total energy possessed of by this little bit of mass M. So that little bit of mass M will have total energy de composed of kinetic and potential added together at any moment in time. We have the expressions for those two things, dk and du. And you'll notice that if you pull out all the multiplicative factors, you'll be left with the same coefficients multiplying a sine term squared and the same coefficients multiplying a cosine term squared. So you wind up being able to pull all those multiplicative factors one-half times and why not?  And there's a trigonometric identity that comes into play. Sine squared plus cosine squared is one. And so the sine and cosine functions vanish from the total energy of this little chunk and all that defines its little bit of energy that it possesses at any moment in time is that the total energy of that little chunk is constant. It may be divided differently between kinetic and potential but the total energy of that chunk of mass that makes up that string that's vibrating is constant and it's given by this number here. And again it depends on angular frequency, linear mass density, the length of the element, the initial displacement of any element of the string and so forth. This is just the energy stored in this little piece of mass M at a location x in space and t in time. Now note that the total energy depends on the square of the angular frequency. The presence of the omega squared multiplier tells us something about the number of time derivatives or the product of the number of time derivatives that had been present in the original equation for total energy. Remember we had to square the time derivative of the wave function that solves this mechanical wave equation and that yielded an omega squared term in all of this. So you can see that there are shades of the number of derivatives left over as sort of vestigial elements of the energy equation for this little bit of mass M. So here are the key takeaways from this look at mechanical waves. The mechanical wave equation relates the second derivative with respect to space and the second derivative with respect to time. You would derive its form in a dedicated class on waves but we don't have time for that here. Nonetheless I want you to take away the big lessons from this. Now the solutions to the wave equation when acted upon by the derivatives in the wave equation yield squares of the angular frequency omega and the wave number k. Recall we had an equation relating omega squared k squared and the speed of the wave squared. The energy equation for the wave or a part of the wave is sensitive to the number of time or space derivatives in the underlying equation and these manifest as multipliers like omega squared. And one might think about the presence of the squares of these quantities like omega or k as indicative of the underlying wave equation that you needed to have solved in order to get these solutions in the first place. Now let's take a quick look at waves and electromagnetism. This is the next classical wave equation that was discovered in the history of physics and it's derived from Maxwell's equations for electric and magnetic fields. So the wave equation that results describes the propagation of oscillating electric and magnetic fields in empty space for instance although it's not limited to only empty space and that wave equation can be written as follows. That the speed of light in empty space squared times a spatial derivative squared minus a time derivative squared all of this acting on an electric field vector is equal to zero. Well again notice that like the mechanical wave equation we've got second derivatives in space and second derivatives in time all acting on a solution E vector whose form we don't necessarily know beforehand. But if we solve the equation we find out that the solutions to the oscillating electric field components look very similar to the mechanical waves in that they have a vector amplitude instead of a scalar amplitude a cosine function a spatial piece and a temporal piece of the argument of the cosine function. Now I should note that yes there is an identical equation for magnetic fields that can be derived from Maxwell's equations but you can think of it as a bit redundant it describes the action of the oscillating magnetic field but can be related through the mathematics of the solution to the electric field and so if you can remember the electric field wave equation which I'm not asking you to do but will come in handy in a dedicated course on electromagnetism later you can very quickly work out what the form of the magnetic field wave equation is and relate the electric field to the magnetic field although they are in independent directions of one another the field strengths are not independent of each other. This is an interesting problem in that electromagnetic waves are two component waves they have an oscillating electric component and an oscillating magnetic component they have two kinds of information that are stored in the wave and this is a theme that we'll return to later when we look at matter waves. Now applying the wave equation written here to the solution written here similarly yields quadratic multipliers of k squared and omega squared so for instance we find that the speed of light squared will be equal to the angular frequency squared divided by the wave number squared and this latter relationship turns out to be a direct consequence of the massless nature of light that we learn from Planck's relationship for energy and momentum and wavelength and so forth and the special relativistic relationship between energy and momentum for massless particles. This allows its speed to be related directly to its frequency and wavelength with no other multiplicative or additive factors involved. So let's revisit that the relationship between frequency and wavelength through a wave can be directly related to the energy present in the radiation quantum the photon. The photon is the particle-like aspect of light's behavior and the electromagnetic wave is the full wave description of light's behavior. So for example from above and from our previous look at electromagnetic waves we know that the speed of light can be related to the wavelength and the frequency of light as follows wavelength lambda times frequency f. Now to get angular quantities shoehorned into this equation like omega and k what we can do is we can multiply lambda f by a clever number one so I'm gonna multiply by 2 pi divided by 2 pi. If I group the 2 pi in the numerator with f I get omega the angular frequency 2 pi f. If I group the 2 pi in the denominator with lambda I get k. k is just 2 pi over lambda. So I wind up with the equation that the speed of light is omega divided by k the angular frequency divided by the wave number. Now recalling that Planck's relationship for the energy and frequency of light related by Planck's constant is e equals hf. We can play that same game and shoehorned angular quantities in here by multiplying hf by a clever number one 2 pi divided by 2 pi. Grouping the 2 pi in the numerator with f the frequency in the numerator and taking h and dividing it by the remaining 2 pi in the denominator we wind up with this compact equation for the energy as related to the angular frequency. h bar is to denote h over 2 pi and it is known as the reduced Planck's constant. Remember that Planck's constant has a value of 6.626 times 10 to the minus 34. So all you do is take that divide by 2 pi. 2 pi is approximately 6 and so you wind up with a number that's about 1 times 10 to the minus 34. It's very convenient for all of these angular wave concepts to carry h bar around rather than h and 2 pi. So it's very nice to define this reduced Planck's constant notationally as an h with a line through the vertical part of the h. Now the momentum of the quantum is given by p equals h over lambda and again if you insert a clever one in that you'll find that this is equal to h bar times k the wave number. What's nice about this is it kind of puts e and p for an electromagnetic wave on equal footing. e is h bar omega, p is h bar k.  At least I find them more convenient once you feel more comfortable with angular frequency and wave number as angular concepts of oscillatory motion. Recalling that the speed of light for our electromagnetic waves is given by omega over k that is the angular frequency divided by the wave number if we substitute in for those two quantities with their energy and momentum expressions we recover the special relativistic relationship for light as a massless phenomenon. e equals p times the speed of light. So we've exactly recovered the Einstein energy relation for light a massless phenomenon. So let's take an overview of the wave equations and classical mechanics and electromagnetism. They involve second derivatives of both space and time. So you see both d squared dx squared and d squared dt squared in these wave equations. Now we could have inferred that from the results of the wave equations as their application results in squares of time and spatial frequencies. So for instance omega squared and k squared appear in equations that result from the application of the wave equations. The energy equations tell us the proportionality of frequency to wavelength as well as other useful information like that. And all of this kind of leaves you wondering if we have a mechanical wave equation and an electromagnetic wave equation where's the matter wave equation? Where's the evidence for that from the history of science up to the early 1900s? That's the problem. Since its presence could not be inferred directly from previous measurements in the same way that Maxwell's equations were inferred from Coulomb's law and Gauss's law and other things like that, Ampere's law, and ultimately when composed together in the form of Maxwell's equations led to the wave equation for light. And in the same sense that considering a string as a bunch of masses bound by springs that are you know tugged up and down and then caused to vibrate by being stretched will lead you to the wave equation where where's the exercise that leads to the wave equation for matter? And that didn't really exist up through the 1920s, the early 1920s. So the question I would put to you is is it possible given other equations that we could infer from what we know its form? So if we know things about particles and waves like wavelength and frequency and energy and momentum and we know the relationships between those things, can we figure out the wave equation using all the information we have from atomic spectra, the black body radiation spectrum, the photoelectric effect, Compton scattering, and all that other stuff? Can we figure that out? So one can glimpse the hints of the underlying but unseen matter wave equation.  And we can get that glimpse of the shadow of the matter wave equation by considering the conservation of energy for a particle that's acted upon by an external force. Now such a particle in classical physics would have of course a kinetic and a potential energy with specific forms depending on the force involved in the problem. Now conservation of energy would require no matter what that the total energy of that particle is going to be the sum of its kinetic and potential pieces. Now sticking to classical physics for a moment because employing special relativity to derive the rules of matter waves involves a whole skill set of mathematics that really can't be expected of you at this stage of a university career. We're going to stick with purely classical low velocity matter. Even small matter like electrons we're going to have to consider moving at relatively low velocities not very close to the speed of light. Now obviously that doesn't cover the full domain of phenomena of small particles like that but it'll get us going and it will allow us to solve a great number of problems that are actually within our grasp once we figure out the matter wave equation. So using classical physics we can write the kinetic energy as one-half mv squared and we're going to leave the potential energy unspecified I'm not going to worry about what the force is that's acting on this let's just say it has a potential energy u for now and leave it at that. Now we have relationships for matter waves between total energy and frequency and total momentum and wavelength but we don't have momentum in this equation so let's get momentum into this equation and the way we do that is we multiply yet again by a clever number one so if we insert a number one in the kinetic energy equation that is just m divided by m then we get an m squared v squared in the numerator and mv is just momentum so we wind up with momentum classical momentum squared in the numerator divided by twice the mass of the particle plus its potential energy u. So we have our kinetic term now expressed in terms of momentum and we still have our potential energy term and they're summed together to get the total energy e. So let us now inject de Broglie's postulates into this equation that is e equals hf which is equal to h bar omega and p equals h over lambda which is just equal to h bar k again employing all these nice angular quantities and if we do this we now obtain the shadow cast on the wall by the matter wave equation and that is h bar omega equals h bar squared k squared over 2m plus u. You see it? The single power of omega on the left side indicates to us that a shadow is cast here by a single time derivative that's acted on some solution to the underlying wave equation. We don't see the solution and we don't see the wave equation but we see the result of applying those two things and that is a single power of omega on the left side. The k squared on the right hand side implies that there's a second derivative with respect to space in the wave equation acting on the solutions to that equation whatever they may be. So we have a single time derivative and a second space derivative that result in k squared and omega. So let's go ahead and take that equation with our hypotheses our hunches about what the underlying wave equations form might look like and let's try inserting those hunches into this equation above assuming that an appropriate derivative has acted on an unknown solution to yield a single omega or a k squared. So if we do that if we take our hypothesis about the number of derivatives acting on some unknown function that solves the wave equation yielding this relationship we wind up with the following equation. On the left h bar times the first time derivative of an unknown solution which I'm denoting with the Greek letter psi and it's a function of space and time we're only considering motion in one dimension right now. On the right hand side we have h bar squared over 2m times the second derivative with respect to space of that same function psi of x and t and of course we have the potential energy of the the matter wave still tacked on to the right hand side over here and I'll return to that a little bit more later. So this looks promising it has all the hallmarks of a wave equation but it's different from Maxwell's equations or a mechanical wave equation in one key way. It has a second derivative in space but only a single or first derivative in time. This will have implications for the kinds of functions that can solve such equations and the solutions to this as I've said are denoted by the capital Greek letter psi as a function of x and t in one dimension. So let's explore solutions to this equation and as we do this we'll find that we are missing at least one key piece of the underlying equation. We've guessed at the form of the object casting the shadow on the wall and we may have guessed incorrectly. So let's begin by guessing the form of the solutions to our equation and then plug them in and see if we recover our energy conservation statement. So to simplify matters let's consider for now free particles that is particles free from external forces and that is most simply expressed by setting u to zero. The particle has no potential energy associated with it we're only considering motion at a constant velocity so that's a fixed kinetic energy which then relates to the total energy of the particle. Okay well to solve our wave equation we need a kind of function that when acted on by a derivative transmutes into another version of itself. So for instance in the old wave equations the mechanical wave equation the electromagnetic wave equation we had second derivatives acting on the solutions. Sines and cosines were great for that because after two derivatives they returned to their original selves. So that's what we did for traditional waves we used sines and cosines. So let's take a guess let's guess that psi of x and t is just one of our mechanical wave solutions a cosine of kx minus omega t. Though we're not doing anything original here we're just taking mechanical waves getting inspired by them and blindly applying that idea here. So let's write down our wave equation that we've guessed. H bar times the first derivative of our solution with respect to time. H bar squared over 2m times the second space derivative of our solution. Let's go ahead and plug our guess at the solution in. All right so we plugged our function in on both sides now and then go ahead and work out the derivatives and you should find the following conclusion that we wind up with a positive sine function on the left now multiplied by omega. We get a minus sign from the derivative of the cosine but we also get a minus sign from the derivative of its argument with respect to time. So this winds up being net positive on the left side but over here the two derivatives of the cosine yield an overall minus sign. So we have a positive sine function and a negative cosine function. We can't cancel out the sines and cosines on either side. It doesn't recover our original energy conservation expression. It doesn't work to solve the equation. The left side and the right sides don't give us what we would have expected based on where we had derived this from which was the conservation of energy. And if you try just a sine function it will similarly fail. So what if instead we combine sine and cosine functions? What if we add together sines and cosines? Because when you take the first derivative of something that's a sine plus a cosine you'll wind up with a sine and a cosine in the result and similarly with the second derivative. Maybe a superposition, an addition of sine and cosine will do the trick. All right let's go ahead and try that. Now I'm trying the barest simplest superposition. I'm assuming that they have the same multiplicative coefficient out in front a whatever that is and otherwise it's a cosine of the same argument and a sine of the same argument all added together. When you're playing around with solving equations whose solutions are not known to you a priori that is with prior knowledge beforehand guesses like this will get you through the process and you should always try to start with the simplest guess and work your way up in complexity. So for instance it may be these coefficients aren't supposed to be the same but don't start by assuming that. Try assuming them and then work your way up to a more general set of solutions as you get more comfortable with solving the problem. So we write down our guess at the wave equation again. We plug in our new choice of the solution. We work through the derivatives and we'll get the following equations. Now what I want to do is I want to reshuffle the term order on the left side. I want to get the cosine first and then the sine second so I'm just going to move these terms around without changing anything about the equation and this is the final form of the equation I get.  I can't cancel these functions out. I can't recover the energy conservation expression we started from. It's a lot closer than we were with just cosine but it's still no good. We've got problems with the plus and minus signs and all of this. It's a mess. What's the real problem we keep running into here? Well we keep generating stray minus signs from the single derivative of only the cosine on the left side. The derivative of the sine gives us something positive but the derivative of the cosine gives us negative sine function and that's what's really hurting us here. Our goal at this point is if we're going to figure this out we've got to find a way to get rid of this minus sign we keep picking up and at this point it helps to remember that there are other kinds of numbers than real numbers in the world. So everything I have done up till now is predicated on the assumption that these solutions and perhaps even the wave equation itself can only be composed from real valued numbers you know like 1.1 or 2.3 or pi or negative 75.6. Those are all numbers that can manifest in the real world. If somebody says look I'm you know I'm going to give you negative 76 dollars it means that they're going to take 76 dollars away from you, right? That has real consequences. Negative numbers are real things in the world around us but there are other kinds of numbers that don't have physical meaning in the world around us and it's important to remember that and they fall under at least one class of these numbers is a category known as imaginary numbers and in particular it's helpful at this point to remember some of the behavior set of the archetypal imaginary number i which effectively serves the role of being the number one in the imaginary number set. So let's pause for a moment and revisit imaginary numbers which presumably you have seen in some context prior to this course. Let's recall that the imaginary number i is a special kind of number one but with no physical interpretation. So i is defined by the question what squared equals negative one and the answer to that is i and i's value would be the square root of negative one which is nonsensical. If somebody told you you know give me i dollars you would not know what to do with that because you don't know how to calculate the square root of a negative number and then turn that into a real dollar value that you then hand to that person. Now this number can exist in a mathematical universe where it has plenty of self-consistent rules that don't violate any of the axioms of mathematics that you're that you're playing with. And in fact i doesn't violate any axioms of mathematics at all so it's perfectly mathematically tenable even if it's not physically realizable around us in the world. Its existence mathematically has consequences though. So for example you know going back to the question that leads to i you can take i and multiply it by itself that's just i squared. If we plug in for i squared we have the square root of negative one times the square root of negative one and by definition that has to yield negative one. And so i squared equals negative one i is the answer to that question that you know the square of what number gives me negative one. But you'll notice that i squared has the ability to add a stray minus sign where none would have been present before. And i is the number one in the imaginary number world. It is the unit on which you can build all other numbers, integers for instance, in imaginary space. So the presence of extraneous minus signs when trying to solve equations using functions as we've been doing with our guests at the matter wave equation could actually be an indicator of something that we are trying to use real valued numbers and solutions only. But maybe the problem we're trying to solve is too complicated to only admit real numbers that it requires the ability to store additional information that real numbers alone cannot accommodate. Those can be accommodated by complex numbers. These are numbers that contain both real and imaginary components. So for instance the complex number z is made from two real numbers x and y, but y is multiplied by the number i. And so i, y is imaginary, x is real. This is a combination of a real and an imaginary part. This is what is known as a complex number. And it stores twice as much information as a single real number because it's got this extra component over here. And if this is reminding you a lot of vectors like a vector z being equal to a component along the x axis x and the y component y, that's good because a lot of the basic ideas of vectors translate into complex numbers and give us some confidence about how we can get useful information out of complex numbers. So let's explore complex solutions to this equation. So let's start by trying a guess at a complex solution. It's got a real valued part, a cosine, and it's got an imaginary part, a i sine. So all I've done is I've added the number i to the sine part of my solution. So again here's my guess at the wave equation. I plug in my solution, I take my derivatives, and I get the following results. Now again I've got sine and cosine out of order on the left side. So if I shuffle them around to get cosine first and sine second and try to map that onto the right, I see that I still have a problem. I've got negative i cosine and negative cosine here. I've got a sine and I've got negative a i sine over here. I can't just naively cancel these things out. That doesn't really work. So I have a problem. I still can't get this to work out. If I were to try to move an i or a negative i from the left side to the right side, I'd still wind up with a stray i where I don't want one. And that isn't going to work for this problem. So it's ridiculously close to working out. We're so temptingly close to solving this problem right now, but something is still missing. Some salt is missing from our soup here that we used to try to mimic the recipe for the matter wave equation. So let's see if we can get one more opportunity to think about our assumptions by moving some minus signs around to see if we can find a clue that will resolve this little puzzle. So I'm starting with the last equation from the previous slide here. I haven't done anything to it yet, but what I'm going to do is I'm going to pull the minus signs on the right hand side out as an overall multiplier and then multiply both sides by negative one. So I'm basically moving the minus sign here to the left side of the equation. So this is the net effect of doing it. Notice all the terms over here on the right are now positive. The minus sign that would have been on the right is now moved to the left and overall multiplies these terms. And if I go ahead and do that, if I multiply it through, I see that I now have i cosine here and negative sign here, and I have cosine here and i sine here. And that has a kind of weird rhythm to it. And that's the clue. The difference between the function on the left hand side and the function on the right hand side boils down simply to a missing factor of negative i on the left. If we had originally guessed that the wave equation was negative i h-bar times a time derivative of the wave function, all of this would work out. We would actually get the left hand side being equal to the right hand side. So if we had just done this at the beginning, if we traded our original guess at the left hand side of the wave equation which was all predicated on our bias for real numbers, h-bar times the first time derivative, if we traded that for negative i h-bar times the first time derivative, then we'd solve our problem. And let me show you that. So to wit, let's revisit our guess at the form of the wave equation. I'm leaving the solution completely intact. That's still the same as what I had a moment ago. And what I've done is I've changed the left hand side of our guess at the wave equation to be negative i h-bar times the first time derivative of psi. I've done nothing to the right hand side. Plug in the solutions. Do the derivatives. Play the game again where we shuffle the terms on the left hand side to get cosine first and sine second. Move the minus sign on the right over to the left and distribute that into these terms. Now multiply the negative i into the function. The negative i multiplied by this i gives me negative times negative one, which is one, positive cosine. The negative i multiplied by the sine function cancels out the minus sign here and leaves me with positive i times sine. Cosine, i sine. Cosine, i sine. The left and the right sides level out and the function is now the same on both sides and it can be cancelled out mutually on both sides of the equation. So, Bazinga as a famous tv character might say at a moment of revelation. We've done it. We've cracked the underlying form of the matter wave equation. Let's prove that by seeing if it returns to us the energy conservation relation that we started from. We've only shown here that we have a wave equation that when acting on our guess at the solution to a free particle wave returns the solution with a bunch of multiplying coefficients and so we can cancel a function out of both sides. But we don't know that we've recovered the conservation of energy equation we started from. So, let's see if this all works out. Let's review what we did. Starting from the total energy of a free particle, which is h bar omega equals h bar squared k squared over 2m, we constructed a wave equation that had the right time and space derivatives in it to return these factors of omega and k squared. We played with oscillatory solutions and we learned that only complex functions will satisfy an equation like this. And from this we inferred a missing imaginary number from our original guess at the equation. We should have had a negative i lurking on the guess of the left side of our equation the whole time in order for this thing to have viable solutions for the free particle that work out. The solution guesses that we made for free particles are of this form. They're complex. They have a real part and an imaginary part. And plugging them into the wave equation yields this relationship. H bar omega equals h bar squared k squared over 2m, which was precisely the energy conservation equation for a free particle that we began with. So, it's entirely a self-consistent exercise. And that should be deeply mathematically satisfying, even if you're not completely comfortable with the process by which we arrived at that. But I promise you that this is not the first time, assuming this is the first time you've ever seen this kind of strategy done for solving an unknown equation with unknown solutions. This is not the first time you'll bust out this trick in your life if you ever have to solve hard problems. So, this trick is actually useful even if it feels a little clunky at first. And I hope it conveys to you the sort of incredible exercise that must have been required originally to derive this wave equation in the 1920s. We're doing this with the benefit of a century of hindsight, but our predecessors did not have the benefit of this much hindsight. So, while I am able to look at resources and come up with ways of explaining the wave equations form to you at this level of a physics course, the people that were involved in trying to write down the matter wave equation in the 1920s did not have the benefit of this much hindsight, and so they were struggling with immense difficulty in a different mathematical landscape than we are in now. So, putting back the potential energy piece of this, this is the full one dimensional what is known as Schrodinger wave equation. We have another function that we can tack on to the right-hand side here, v of x and t, which is known as the potential term. That function acting on the wave function psi would return the potential energy of the matter wave, in this case u. And this is one of the most important equations in history where I've added back the potential energy term to complete the equation for a particle in one dimension. The one-dimensional Schrodinger wave equation is one of the most revolutionary insights into the universe in the history of our species, named after Ervin Schrodinger, the first person in 1926 to fully determine the form of the matter wave equation. Now, he was doing this using a whole different set of mathematical algebraic and calculus constraints. That is, this is not how he derived this equation, but this is sufficient at this level to motivate where an equation like this might come from. He was using mathematical guidelines to infer the nature of the equation that we simply don't have the mathematical foundation for at this stage of a physics course to do.  Okay, that might seem daunting to you, and perhaps it is. But there's actually a much more difficult part to this equation, and that is that the hard part of the Schrodinger wave equation is the solving of the equation to specific situations. The real challenge of this equation is not this equation itself, although it doesn't look very pleasant, I know. But rather in the finding of solutions, the size of x and t to this equation given different potentials, v, and so forth. Now, we've effectively solved the free particle case, and we'll explore the solutions to that for the rest of this lecture. But if a potential energy term is present, so in other words, if there are forces in a problem, and you can't ignore them, and you have to include them, and those result in potential energy or changes in potential energy for a particle in a problem, you have to completely rework this equation and find the correct solutions that satisfy the equation with the correct potential term added on. And that is much more difficult, and that is effectively what we'll spend the rest of this course learning how to do for different situations that map on to the physical world. Now, I should say that solving this equation for different situations is what has allowed us to understand semiconductors. It is part of what allowed the revolution in microelectronics to happen in the first place. Solving this equation for atomic system leads to an understanding of where chemistry comes from, and specifically doing this on a grand scale is the heart of physical chemistry. Solving this equation for information systems is what results in quantum computing and quantum information, which is a hot subject these days and is one of the many technological frontiers of our species. I cannot understate how important the Schrodinger wave equation is for all the foundations of the world we live in today, but also all the potential for the great discoveries of tomorrow.


In his psychology classes, Jeff often heard terms such as review the literature or literature review. At first, he was a little confused by these terms. If he needed to do a literature review, did he need to find novels with psychology-related topics? Jeff soon learned that, in the academic world, the term literature is often used to describe the collected body of scholarly works related to a topic. In most cases, the literature consists of scholarly books or articles published in academic peer-reviewed journals. So, when Jeff was told he needed to review the literature, this meant that he needed to find previously published scholarly sources relevant to his topic. Jeff learned that there were several reasons to conduct a literature review. Looking at previously published sources helped him to increase his own knowledge on the topic. It helped to keep him up-to-date on current trends and major findings related to his interests. He was also able to identify any areas of agreement or disagreement among scholars related to his topic, and it helped him to identify directions for future research by allowing him to see the problems or questions related to his topic that needed additional research. Jeff found that he was often required to include a section in his research papers where he summarized the previous literature on his topic and showed how it was connected to his own research. Jeff also learned that, in some cases, the term literature review is used to refer to a specific type of scholarly article. In a literature review article, the author does not conduct their own research study. Instead, the author summarizes and analyzes previous research on the topic. For example, the author might review 20 research articles on a topic in order to determine what conclusions can be drawn by combining the findings of all of them together. To sum up, in the academic world, literature is a term often used to refer to all of the previous scholarly research on a topic. Research papers usually include a literature review section, and in some cases an author may publish an article that is only a literature review.