Good morning and welcome. Thank you for connecting to this morning's class. Let's pray and get into God's Word. We'll study from Hebrews chapter 12 today. Let me begin with a word of prayer. Heavenly Father, we thank You for the opportunity to engage in Your Word. Father, we pray that even as we go through this passage, Lord, from Hebrews 12, that You will speak to our hearts. Lord, we ask for Your blessing upon all of us, Lord, students, faculty, and our families. Lord, continue to strengthen us in You, Lord. We commit ourselves into Your hands. In Jesus' name, we pray. Amen. Let's go straight to Hebrews chapter 12 here and we can begin to read it. It's probably a familiar chapter for a lot of us. It carries, you know, some difficult themes and some fairly easy instructions so we can go through it. We can read part by part and that should be alright. So let's first begin with the initial section here. Let's read from verses 1 to 5 and then we will move on to the other section. Hebrews 12 verses 1 to 5. Therefore, we also, since we are surrounded by so great a cloud of witnesses, let us lay aside every weight and the sin which so easily ensay us, and let us run with endurance the race that is set before us. Looking unto Jesus, the author and finisher of our faith, who for the joy that was set before Him endured the cross despising the same and has sat down at the right hand of the throne of God. For consider Him who endured such hostility from sinners against Himself, rest you, become weary and discouraged in your souls. You have not yet resisted to lordset striving against sin and you have forgotten the exhortation which speaks to you as to sons. My son do not despise the justening of the Lord nor be discouraged when you are rebuked by Him. Thank you. Thank you, Super Shish. Let's go ahead and look at what this means. So far we saw how there were the lives of several men and women of faith enlisted by the writer of the Hebrews with the intention of encouraging believers that faith will help them overcome any difficulty or challenge that they may be going through and that they can hold on, they can wait upon the Lord and receive the promises that God has made towards them and right now He flows in the same thought. What is that thought? Don't be discouraged, hold on. Faith will cause you to walk into the promise. So with that thought in mind, He's talking to them about keeping on, keeping on, if you want to call it that way. So to not give up basically. So He says, we are surrounded by so great a cloud of witnesses. Who are these witnesses? It's like a picture of us running if we can look at, consider a race or a sports event, excuse me, where athletes are running. We usually have spectators who are cheering away those who are running well, those who are almost at the finish line. So these people who are cheering are the cloud of witnesses, witnesses to the event that is unfolding before their eyes. The event which is unfolding in this particular case is our lives and our walk of faith. So who are the witnesses? Who are the witnesses that He's speaking about? Any idea, any guesses? The witnesses being talked about are those people who already appear in the space through all of them? Yes. Abraham, Isaac, Jacob, Joseph, among others. Thank you. Yes. Thank you. Thank you so much, Lubega. So that's right. Because we have already spoken about these people who have conquered by faith, it's as if they are now looking at us to finish our race and they are sitting in the stands and cheering us away and they want to see us reach the finish line. They want to see us complete what God has called us to do. So these are the great cloud of witnesses. Some commentators say that you know the people who have passed on, those who are in heaven are probably the cloud of witnesses. However, when we read in continuation and also because there's that word therefore it's continuing from the passage that has all these names. So our understanding is it's these people whom we have spoken about. They are watching us finish our race. It says because now they are standing to see how we will do in our journey of faith. He gives us some keys how to finish this race. One key, let us lay aside every weight and sin which so easily ensnares us and let us run with endurance the race that is set before us. So he talks about laying aside every weight. What are some of these weights that we carry during our journey of faith? It can be as you know Jesus talked about the things that choke the growth of the word. He said the deceitfulness of riches, the cares of this world. So there are many such things that we carry and it could become a burden in our in our race. It could be discouragement. It could be disappointment. We've spoken about all these matters. So it can be anything that is heavy on us and it can actually slow us down. We all know that even in the world of sports today the athletes who want to win usually will go to go as far as excuse me wearing clothes that are light or having gears that are so light because they don't want anything to stop their time. They don't want to compromise on their time of completing on the record time that they you don't want to make it in and so if athletes, earthly athletes can be so very particular about running light how much more us and that's the point that the writer is making here. He's saying may our lives be light. If there's anything which is a distraction, if there's anything which will slow us down, let's get rid of it and then he introduces this thought of and the sin which so easily ensnares us. Now obviously sin has a huge part to play in derailing us from the purpose of God because he has stated here that sin ensnares. What is ensnares? Ensnares is like entangles us. We don't know how to get out of it and we've understood the nature of sin. It promises us more than what it offers and usually people get in hoping for a lot more and then the next thing they know is they are caught and they are sucked in little by little, little by little. That's ensnaring. That's entangling, right? Slowly it kind of binds us and then before you know it, you are so stuck that you don't know how to free yourself. So this is the way sin works in anyone's life and he's warning the believer. If we get entangled in sin and snared in sin, it's so difficult for us to come out. Now even if we do come out, you see in the race that we want to win, we are slowing ourselves down. So why give place for sin? Why give place for any form of fleshly desire, any form of you know, a fleshly investment from our side? Let's not have anything to do with those things because in a race, all these things slow us down. He said, first every weight, it can be anything that we ourselves evaluate and say, hey, this is slowing me down. You know, lack of discipline is slowing me down or lack of vision is slowing me down or the company I keep is slowing me down or the lack of knowledge is slowing me down. We are the best judge for ourselves. Maybe in this given season or in general in our lives, what are those weights that we can identify? We've got to get rid of it so that we can run fast. We can run with purpose intentionally and sin, sin of course, right? Because it will slow us down. We'll end up wasting time getting into sin, getting out of sin. And it also has consequences. It causes so much destruction in our lives. So as ministers of God, the weaknesses that we identify in ourselves, we've we've got to be intentional about guarding ourselves in those very areas because we don't want to slip.  And let's move on. He says, let us run with endurance, the race that is set before us. So when we look at that word race over there, the Greek word is Agona. We all know the English word Agony. So that has to do with struggle, agony, right? Struggle, conflict of many kinds. So running the race, it's not so easy. He calls it Agona, which means that there are going to be challenges, there are going to be hindrances, there are going to be difficulties, oppositions, you know, lack, struggles of every kind that we may encounter as we are running this race in order to finish it. Now when we are running this race, he introduces this thought of endurance, run with endurance, endurance the Greek word is a Hupomon and it means having patience, having patience. So we need a capacity which is, which enables us to be patient while we are still away from the finish line. And we've again talked about all these matters, how a delay in the fulfillment of the promise can be so disheartening and that also can slow us down, that can also discourage us from running the race. So a delay is quite challenging or let's say and some uncertainty happened. We never expected life to go a certain direction and that happened that also can cause us to stop running. But in all these circumstances is where we need endurance, the patience to believe God that we will make it to the finish line even though we are going through some of these struggles. So endurance can also be explained as, explained as bearing up under pressure. See it's easy to run when there's no pressure, we can make it, it's easy. But when there is pressure, when there are difficulties, you know, when it is so, it takes more on your part to have the will to keep doing it. That's when endurance kicks in and it should kick in for us who want to reach the finish line. And that's what he's talking about. He's saying when the going gets tough, what do we need at that point? Endurance, some patience where we have faith in our heart saying that okay, if I keep going, I am going to see the results. I am going to reap the rewards. Now while we are enduring, it's not easy. There can be several battles, especially in our minds that convince us to give up. So he's giving us another key. First he said, come on, run light. Okay, run in such a way that sin cannot get you, guard yourself. Then he says, okay, run with endurance.  You're running really well. And then he gives another key to make it even better. What is that key? He says, in your mind, focus on Jesus. Okay, focus on Jesus while running the race. So he starts in verse 2, looking unto Jesus. Who is Jesus? The author, the finisher of our faith. So his life is an example for us. He has lived a life of faith and he's the one who's the beginning of faith, who's the end of faith, who has applied faith in his own life and he has completed the mandate that God had for him. So if I want to get to the finish line, here is a key. Look unto Jesus. Focus on Jesus. Think about Jesus. Look at the life of Jesus as a pattern and observe it, apply it in our own lives. How did he make it? Was his life easy? Was his life very convenient? Was it smooth and simple that he could just, you know, reach the finish line and say, okay, Father, it is finished. I have completed the mandate that you gave me. No, we all know there were ups and downs. We all know that he too faced struggles. But how did he do it? This is the way Jesus did it. It says, who for the joy that was said before him endured the cross, despising the shame and sat down at the right hand of the throne of God. So here is a tip or, you know, here is a method that you and I can use to make it to the finish line even if the journey is hard. And that is to always keep the goal or the reward in mind. Okay, so whenever we have the goal in mind, the struggle doesn't seem so difficult. I mean, I'm just using one simple example. These days people are all talking about fitness and losing weight and, you know, building muscle and all. Now, if they have a goal like, okay, I want to lose this much weight. I want to, whenever people are motivated, what happens? They're ready to do because they're imagining themselves fit and, you know, healthy and active. So that motivates them. So that images in their minds and to get to that place, they know that today they have to make some sacrifices. They have to, you know, be strong and they have to push themselves. Even if they don't like doing what they are doing for their own health, they're able to do it, right? Or think about a student who wants to crack the exams or crack a competitive exam in their minds. They're thinking, okay, you know, I will have this position. I will have this option to study in this university. Is it easy to put in those many hours? Is it easy to be disciplined and make time to study? It's not easy. It's definitely not easy. But what did Jesus do? This is exactly what Jesus to did. He looked at the joy which was set before him. Okay, and because he was thinking about what was on the other side of the cross, maybe it made it a little easier in the mind for him to endure the cross. So what was the reward that Jesus was waiting for? Jesus was waiting to, you know, be able to provide that redemption to the people that brought him joy. Jesus was waiting to finish the work and go back and see the father. Jesus was waiting to hear from the father that, okay, well done, good and faithful, you know, son of mine. Jesus was waiting to be received in the arms of the father. So these were the pleasures that motivated him. And as he was looking at these things, okay, I'm going to be a blessing to the people and I'm going to be back with my father and, you know, be back in heaven. These were all things that could have given him that joy that, okay, it's quite hard for me right now, but I'm going to end here. So these are the keys that the author is sharing with us and it also states, despising the shame. So was there a shame associated with the cross? There was. Okay, is there a shame? It could be a shame associated with our struggles, the opposition that we are facing. Yes, there could be, but when we are considering the results and the blessings much more than what pain we are going through now, it makes it somewhat easier for us to walk that path and to actually reach the finish line. Okay, so when we are running the race, don't you think it's important to reach the finish line? Yes, that's that's the reason we are all running. It's in in a like a World Cup or you know, like Olympics. Would any athlete come with the intention of I'm just going to have a good time, you know, I'll just run and see I just want to feel I want to feel how it is to run on on that track or to hear the cheers of no athlete who's competing in you know, a championship goes there without the intention of winning. Okay, and so it's devastating when you don't get to the finish line and you know, it's devastating if you don't get that gold medal. Okay, people put in yours their lifetime to be able to get that gold medal and that is the way we as believers have to run our race and we are being invited to run our race. Okay, run like you have to get the medal. Don't just run for a you know, for a fun time or for pleasure. No, that's not what our lives journey is about. But we look on to Jesus. Okay, don't get entangled in the things of this world. Be like a soldier. That's another picture that we see in scripture as far as our Christian life is concerned here. It's like an athlete. But we've also seen other places where it says we are like a soldier. We can't afford to get entangled in the world's affairs because we've got a medal to win. We've got a finish line to reach Jesus had that mindset for him. My father's business. I'm here to do the will of the father. That was the joy that Jesus had and and he always wanted to rejoice in that to be able to serve the father the way that he did. And so for you and me as a child of God, it's the same thing. This life God has given it is for the Lord. We have to do what God has called us to do. I've got to get to the finish line and I don't want anything to distract me. Okay, so in this scenario, we understand discouragement was the big thing, right? So the others speaking to them in many ways and saying, don't let discouragement get you if Jesus did it and now he's sitting at the right hand of the father having completed, you too can do it. So be encouraged and verse three for consider him who endured such hostility from sinners against himself, lest you become weary and discouraged in your souls. I don't know how many of you have seen. There's one picture that usually gets circulated where you know, someone's telling Jesus their problems, but then Jesus is sitting there bleeding. Okay, and you know the way he and your pain.  And we could possibly never even match up to what he has endured, right? And so he already knows and so when we are going through some opposition, we can think about Jesus. He went through more than me, but he didn't give up. So why should I give up? You know, I too can deal with what is happening right now and I shouldn't be weary. My soul should not be discouraged. Verse four, you have not yet resisted to blood shed striving against sin. So he is also giving this comparison and he's saying that we've not even, you know, kind of reached the high point of that test yet. Okay, but Jesus went through it all. He reached levels in his struggle against sin and his endurance that we can always gain a lesson from Jesus. If he could do it, you know, we are nowhere close to him in our struggle and most certainly we can do it. So don't be weary. Don't be discouraged in your soul. Verse five, he says, and you have forgotten the exhortation which speaks to you as to sons. So this is the next section actually verse five and we'll now read that next section and understand it from verse five to verse seven. So it's so precious.  So verse five to verse seven. But in the meantime, okay, we have a question. Yes, Jafina. I mean, I just don't want to mix up the questions. Yeah. So when we say Jesus is the author and the finisher of the faith, what do we actually mean by does it mean that our faith starts the moment we start believing in Jesus? And I mean, I just want to know that what the phrase actually means. So author and finisher of our faith could mean I just looked at my notes. I haven't commented on it. But my thoughts are that our faith begins with Jesus. Isn't it? Because he is the word and our faith begins with him and our faith author and finisher. I don't think it's even referring to the end of the faith. Like when it says finisher, but it could mean that he will help us complete our journey of faith. So our faith begins with him and he is the one who will also give us the faith to complete our journey. So that's my take on it. If there are any other any of our friends here who also want to share your view, please do. Is that okay, Jafina? Okay, great. And also the last thing that we did in verse four, you have not yet resisted to bloodshed again since so it just means the cross. I'm at it. Thank you. Yes, even not just the cross, but if you consider the Garden of Gethsemane even then there was this temptation to give up and we know that Jesus was sweating blood. We know there was an internal mental and spiritual battle, but he fought it in such a way that he won over it and he didn't give up. He still went to the cross. So we could even consider that. Yes. Okay, so now moving on to the next section here versus five to versus verse seven. Could somebody please pick it up and actually you could probably read to verse 11 because it has the same theme. Verse five and you have forgotten the exhortation which speaks to you as to sons. My son do not despise the chastening of the Lord nor be discouraged when you are rebuked by him for whom the Lord loves he chases and scourges every son whom he receives. If you and your chastening God deals with you as with sons for what son is there whom a father does not chase but if you are without chastening of which all have become partakers then you are illegitimate and not sons. Furthermore, we have had human fathers who corrected us and we paid them respect. Shall we not much more readily be in subjection to the father of spirits and live for they indeed for a few days chastened us as seen best to them but he for our prophet that we may be partakers of his holiness. Now no chastening seems to be joyful for the present but painful. Nevertheless, afterward it yields the peaceable fruit of righteousness to those who have been trained by it. Amen. Amen. Thank you Rosalind. So this is like the this is some light on the parenting process. So far he said don't be discouraged keep the focus keep running even when things get difficult because we've got to make it to the finish line and now he is reflecting on some of the struggles not all some of the struggles as the chastening of the Lord chastening has to do with correction the kind of correction that parents bring to the children. Chastisement if you look up the Greek word it means education training nurture. So it's a positive word chastening doesn't mean that you know God crushes our soul God causes us to go through so much that we are broken and we are not able to rise above that particular situation that is not chastening. Chastening is like you know God for the sake of education and training nurturing main put in some tests in our way but the what is the goal of the test it is to pass isn't it? So imagine think of this first grader is learning mathematics and they are just about learning you know they've gone till 50 let's say they add up 20 plus 30 they are able to do it and suddenly the parents says okay now tell me what is what is 30 plus 30 now that is 60 and for the child it's a stretch you're like oh I'm being tested in this moment but the what is the intention of the test to be able to help the child go to the next level so any good parent would do that to educate the child train the child build up the child so that is the chastening of the Lord the chastening of the Lord is not that you know God brings judgment and you know he that the believers are so under the judgment that that's it their lives are over that's not chastening because sometimes we look at circumstances around us and we interpret it like that when people go through difficulties we say God is chastening you know God is judging you that's why you're going through this but the tests of the Lord are like the parent it's meant to educate us so every test is for promotion I heard somebody say that and I believe you know it really makes sense because that's how parenting works as well every challenge that a parent poses to the child is not that you know your child will never learn that subject or never learn that life lesson but it's to make a child champion now if the child can stretch beyond that level they go to the next level and then you know there are challenges over there so that is chastisement so with that in mind the author is actually telling see some of the difficulties that we are going through could be the chastening of the Lord so he says verse 5 you've forgotten the exhortation which speaks to you as sons my son do not despise the chastening of the Lord nor be discouraged when you are rebuke by him so there are times when God will bring chastening when he may bring correction but it's all well-intentioned to build us up okay and he says verse 6 for whom the Lord loves he chastens and scourges every son whom he receives so what kind of a relationship do we have with our with our God he's our Abba father and that is why he is chastening us he's rebuke us we we see it also says gorgeous every son meaning like you get a whack or you get a beating sometimes but that's good isn't it and he goes on to talk about like bad parenting like if imagine a parent does not chasing their children that child will grow up to be a rogue because nobody can tell them what to do nobody can correct them nobody can instruct them but a parent who does their due duty of training the child in this manner will raise up a champion okay and that's the way God is working in all of our lives so he says in verse 7 if you end your chastening God deals with you as with sons for what son is there whom a father does not chase it that's not good parenting there's no good father who doesn't chase in their son or daughter so he says look when parents do that we respect them right we go back and we reflect and we say oh thank God you know my parents taught us how to be disciplined thank God my parents taught us you know how to value time thank God my parents taught us how to invest the money correctly or use the money correctly now we thank we honor our earthly parents for some of these practical life lessons that we need in our life here on earth so how much more should we be thankful to our heavenly father for chasing us for rebuking us for correcting us when we are wrong so that we can become stronger and we can become you know partakers of the blessings and the goodness of our God and the most important thing for us to remember is our relationship it's because God considers us that his children that he's doing it now if he's also explaining here in verse 11 okay where was that I think it's probably later on please give me a moment seem to have yeah verse 8 end of it where he says like if a father does not chase it then you are illegitimate and not sons that's a very strong statement because what he's trying to tell us is you know if God where to be careless about us then it shows that he's not giving us the father's interest because you know if if let's say there's a parent and their own child is is misbehaving being disobedient they sharply correct their child because they want them to grow well now if they find another person's child misbehaving they lack the rights to correct that child you know at the least they may try and say something basic but they lack the rights they can't spank the other person's child or in this case you know even a very strong statement is being made where it says that we are not God's children only if we don't have if we don't experience chasening or a rebuke or a you know a scotch so when we go through some of these difficulties where it seems like God is correcting us be open-minded about it okay receive what God is doing in our lives it reminds me of John 15 where we learn about the pruning okay pruning is never easy imagine if we were I mean as the plant when a plant is being trimmed and some edges are being cut it's painful it's surgery and we don't like that we don't want edges to be you know sort of chipped off and cut away but God does that so that we can be shaped for what in John 15 we learn the whole intention of the Godhead the father the son the Holy Spirit they're working in our lives to make us fruitful why are they pruning so that the branch will be fruitful same applies here when God is chasening us when he is working on our lives he wants us to become more fruitful and there is an emphasis on the on the way he looks at us because we are sons he can't let us just be the way that we are so how does he bring this correction okay how does he primarily bring this correction we know it comes from the word most of the time the word speaks to us the word convicts us and as we respond to the word change comes into our lives but of course you know there are other ways as well through which the Lord may want to correct us so we need to identify that so this whole section is about chasening in our lives and the fact that we must yield to it and when we do that we will like those sons and daughters who give credit to their parents for inculcating good habits in them we will praise God for what he is doing in our lives so verse 11 he says now no chasening seems to be joyful for the present but painful nevertheless afterward it yields the peaceable fruit of righteousness to those who have been trained by it so let's go through the Lord's chasening it is only going to build us up it's basically training you know like how we saw athletes so who are we we are spiritual athletes and God will put us through some training you know you got to wake up 5 a.m. and do your warm-ups and run spiritually I'm talking about that so as we are doing that we will be shaped into the kind of people that he wants so we must not let some of these struggles become a discouragement so verse 12 and 13 anything about chasening anything about receiving the Lord's correction all right so as far as chasening is concerned that the point is be submissive so that the process can be quick now moving on verses 12 and 13 so I let me read that it's just two verses says therefore strengthen the hands which hang down and feeble knees and make straight parts for your feet so that what is lame may not be dislocated but rather be healed so the whole point is readiness so because of all these things we've got to finish the race God is working in our lives for fruitfulness for our benefit come on get up now okay all the weakness all the tiredness we usually say right like okay get ready roll up your sleeves so it's a position of readiness don't be fallen and discouraged when we are let's say encouraging a sibling or somebody at home they're so discouraged about the matter we might say come on let's let's nothing will happen it'll all work out let's go out for a walk you know go wash your face put on some good clothes so it's like saying that where he's saying strengthen the hands which hang down and the feeble knees make straight parts for your feet so that what is lame may not be dislocated but rather be healed so give place through your obedience and submission give place for the Lord to bring healing we may feel like you know he's talking about dislocation and all that so when the chastening is going on it may feel painful that something is breaking something is broken but when we heal to the Lord healing will come and that's the faith that one has to keep healing will come so I'll keep going and God is going doing a good thing in my life life now versus 14 through 17 a few more instructions that he has for the believers could somebody read the section 14 to 17 versus 14 to 17 was you pleased with all people and holiness without which no one will see the Lord looking carefully lest anyone fall short of the grace of God that's that lest any root of bitterness bringing up cause trouble and by this many become defiled let's there be any fornicator or profane person like issue for one more is the foot solid his birthright sold his birthright for you know that afterward when he wanted to inherit the blessing he was rejected for he found no place for repentance though he sorted diligently with tears wow these few verses are so loaded I was thinking we can finish in five minutes but let's see you know how far we get so it's starting with verse 14 and this verse 14 is so powerful in all our relationships it's worth remembering this verse 14 it says pursue peace with all people it's so easy to get along with some people but it's so difficult to get along with some other people but no matter what kind of people are there in our circles here is a is a key instruction for us pursue peace with all people the people who are easy the people who are difficult to maintain peace with all the people and you know that's that's very helpful because sometimes we just feel like you know giving it back and making the situation worse and just starting up a quarrel and making the conflict bigger but what does the Bible say it says you know as far as possible pursue peace with all people whoever they are you do your best to be peaceful don't don't make the conflict you know bigger and greater and it says holiness peace pursuing peace is one important thing now coming to holiness it says let's be holy what is holiness holiness is you know maintaining that purity before the Lord and not being tainted by sin by you know fleshly matters coming coming with that cleaness of heart before the Lord and maintaining that throughout our journey right we begin seeking and pursuing holiness as new believers in the Lord and that hunger should always remain even before up until the Lord calls us home so holiness maintaining peace having holiness okay it's easier said than done all these matters but we know that we have got to pursue it and God is calling us to that and verse 15 he says looking carefully list anyone fall short of the grace of God there's any root of bitterness spring up cause trouble and by this many become defiled so now the third point is bitterness where does bitterness happen it happens in our hearts okay again in our relationships we may have had certain experiences that causes to dislike people now if we have forgiven them that's the path that Jesus has taught us to forgive others to let it go and you know the right way so when we forgive the right way we've released it we've released them but when we don't forgive the right way we're saying they're forgiven but then what happens we carry a route in us we understand this isn't it when people are gardening they they want to remove certain plants and then plant something else it's usually advised to kind of go a little deeper and get rid of the roots of the earlier plant or you know whatever was was growing there because unless you do that what will happen that other plant also will start to come up you may have put new seeds and you're excited about all the new things that are happening in life but you see there are some old roots because we didn't go deep enough in our working on ourselves and one of the bad roots is bitterness we've not dealt if you can think of people you know whom we so dislike that our prayers for them sound different you know and we are contemplating about doing something evil saying something evil there's certainly is a root still within us a root of bitterness and we are being warned because it's very dangerous a root of bitterness think about this and we are being warned about the root itself not even about the tree tree is not even come but he's saying even if a root is there it's going to cause so much trouble spring up because it will cause trouble by this many become defiled and he's saying it will cause problem in the church it will cause problem in the fellowship you know among God's people so make sure even the root of bitterness that it is removed and give it more please so I think we had time only for that three things first you peace holiness and get rid of the root of bitterness okay we look at the other other things in the next class so could somebody please lead us in prayer before we close our class today. Lord we thank you for the lecture we've been having Lord again we thank you for the gift of life we thank you for all that we've had from our dear lecturer but I also pray that I also pray that you bless her and her family bless all the kids all the students who are in this call and Lord even those who have missed Lord let them get time to get this on the E platform and also revise Lord we do pray and believe that everything is going to be fine in the name of the Father the Son the Holy Spirit and we say amen. Amen thank you. Thank you thank you success. Thank you everyone. God bless you have an amazing day. Take care.


on the world to study turbulent mixing. So this is the second talk on work we're doing on the NIF laser to study turbulent hydrodynamics in the presence of high temperatures and high radiation temperatures. This we all know, but this is just showing instability on the left hand, on your right hand side, in a classical sense, in the absence of magnetic fields and radiation and the basic equations are being written down on your left hand side. And so this is the starting point. That's the phenomenon we're going to study, but we're going to do it in an environment that's very unique in its own right. And then try and see if we can push forward in some new areas of science combining these two effects. The turbulent mixing hydrodynamics combine with the high energy density setting with lots of high temperature plasma and radiation. That's just stopping it so you can take a look at it. And you can see lots of structures and ripples and perturbations growing, and they evolve with time. And so these are the different types of HED facilities that are widely used around the world. There's several to many. The NIF laser is only one of many of these. And so you can see up here the GECO laser, the Shangguan laser, GECO in Japan, Shangguan in China, the Orion laser in the UK, the LMJ laser in France, NIF in the United States, and the Vulcan in the UK again. And so the field is evolving along with the facilities. So this is a capsule implosion simulation for NIF experiments. And so in the NIF setting, a radiation drive is what forces the capsule to implode inwards. And so the capsule implosion is sort of the simple part of this. The part that the eye catches is all the structures and perturbations that are growing that make the sphere what's supposed to be a spherical implosion very complicated. And so a lot of this leads to hydrodynamic instabilities growing that drive material from the outside down into the very center. And what's supposed to be in the center is a hotspot, which is just something like the center of the sun. But when you inject this outside material of plastic or doped plastic down into the hotspot, it radiates the heat away. And the hotspot doesn't stay hot. And so the nuclear yield is what goes down because the temperature drops. So we have in the ICF program in inertial confinement fusion program, we have a very strong interest in understanding it at the fundamental level how these instabilities grow, how large they get, and how much material they inject down into the hotspot that lowers the temperature when we're trying to make the temperature high and keep it high for as long as we can. I mean, you want high temperature and you want to hold it there for as long as you can to get the most nuclear reactions to happen, the nuclear yield. Okay, so let's see. Just grabbing a couple of snapshots from experiments that examine this, this is an implosion experiment. Notice the time scales are varying by 100 picosecond steps. So 22.3, 22.4, 22.5, and 22.6. You can see the hotspot is changing in time as material from the outside is being injected in. And so these hotspots are plastic material from the outer shell that has been injected into the hotspot, which is supposed to be just plasma, DT plasma, deuterium and tritium. And so when you inject CH dope with silicon or CH dope with germanium down into that hot central region, it cools the hotspot because it radiates. And when you cool the hotspot, the D and T, the deuterium and tritium stop fusing. And so we would like these hotspots and the mixing of the plastic down into the hotspot, we'd like to minimize that because then the nuclear yield would go up and the burn duration would increase. So that's what we're working on. Now let's see, I think I want to back up a couple. That's the, and so here that is the consequence. And so this is, these first few paragraphs are sort of the summary of the talk. The consequence of inserting shell material down into the hotspot, which is quantified in terms of nanograms of plastic. The plastic is the outer shell, plastic dope with silicon. So the amount of plastic dope, silicon dope plastic that makes it into the hotspot in nanograms, when that increases to the right, you can see that the nuclear yield is dropping and this is a long plot. And so when you put plastic down into the hotspot, it radiates the heat away, the DT plasma that's supposed to be hot cools and the nuclear yield drops. And so we're studying this effect carefully and quantifying it so that we can control how much mixing happens into the hotspot so we can not lower the nuclear yield. And we measure the amount, or we infer the amount of mass mixed into the hotspot in terms of the x-rays they emit. The hotspot is supposed to be just deuterium and tritium, so hydrogen isotopes. When you put plastic down in, or especially plastic dope with silicon, suddenly the amount of electrons that can radiate goes up and so the x-rays coming out of the hotspot increase. And so we go from a measured enhancement in the x-ray output to inferring the mixed mass and once we know the mixed mass, we then know how the implosion is being affected by that mixing into the hotspot. And so again, we measure the hotspots there, the x-ray enhancement factor are these bright spots, that's the added radiation coming out of the hotspot because plastic has been injected into it. Okay, so now going back, and so that's the summary of the talk, going back a few years back, how do we make the measurements? The measurements have been under development for several decades now. And so this is going back to the NOVA laser where we were measuring perturbations growing on planar samples as a function of wavelength in an x-ray environment. And so in this case at the Livermore, we use the indirect drive approach which takes the laser, the large lasers, they go inside a cylindrical gold hole rub, it's just a cavity, they convert the x-rays, the soft x-rays then are what drive the physics experiment. And so then we plot the growth factor versus wavelength for pre-imposed ripples on a planar sample on the NOVA laser, that methodology was worked out back in the late 1980s and early 1990s. On the omega laser, we then have the opportunity to explore direct illumination of the lasers on the physics package as opposed to first converting to x-rays if you just directly illuminate the package with the lasers directly, you get more energy onto the sample, but you then have to contend with the fact that lasers are not perfectly uniform, they have hotspots. And so these hotspots imprint upon the sample where you start out, let's say with a flat sample, now you have perturbations formed and those perturbations grow and we try and diagnose them to see how large did they grow and can we find means to understand that growth and control it. Another example of just direct laser illumination driving a physics sample and what the radiographs show that the perturbations that they impart are getting larger with time and larger in amplitude and larger in wavelength. You can see they're evolving to larger characteristic wavelengths and the ripple amplitudes are getting larger and it eventually will shred or break a target into pieces. So that's the effect that we're trying to quantify and then ultimately control. Okay, going to NIF now. So using the NIF laser, we're switching back to indirect drive. So the lasers come into a radiation cavity, a gold, hollow gold cylinder. They convert to x-rays and the x-rays in this case drive a planar rippled foil, the physics package, and then we shine lasers onto a backlighter and take radiographs from an external x-ray source through the physics package and so these time-resolved in-flight radiographs are how we infer or deduce what's happening as a function of time and then we analyze these in Fourier space and you can see that the ripples are growing with and the ripples are growing in amplitude and their characteristic wavelength is getting larger with time. And so those are the kinds of measurements we're making. Here is a capsule and the way it's mounted up in the radiation cavity for NIF and so the outer part is the cylinder that's gold. The lasers come to the inside of the cylinder and convert to x-rays. A spherical capsule that's going to implode is held in the center but you have to hold the capsule somehow the capsule is held with a thin form var plastic web that's about 70 nanometers thick and so we hold the capsule in place with the thinnest foil that we can manufacture so that it's just barely being held there and you'd like the web, the tent, to be very thin so that it doesn't leave large imprint on the capsule because when the x-rays fill up the whole room they blow up the tent the tent creates a pressure perturbation that imparts a scar on the capsule implosion and then as the capsule implodes that perturbation gets larger and it's a large source of material of plastic that gets inserted down into the hot spot cooling the implosion hot spot. So this is some of the other things that can happen in such a setting and so there's two things to look at here one on the right hand side since it's a movie is what we call meteors it's a fill tube or other discreet perturbation on the capsule shell when you implode the shell you can see this is in action this is a, these are experimental x-ray radiographs I'm not sorry, x-ray emission so we're measuring the hot spot in x-ray emission with pinhole cameras and you can see a perturbation that existed likely from the fill tube is launched as it enters the hot spot you can see as the x-ray yield is coming up when the perturbation gets to the center it radiates the heat away and the temperature drops and the burn stops so it's a very good demonstration of a large perturbation once it gets to the hot spot that's supposed to be very hot it starts to radiate the heat away and the temperature drops and the yield stops and the kinds of perturbation patterns that evolve from a tube a fill tube for example are a little bit surprising the fill tube is a tiny little sort of 15 micro, 15 millimeter diameter tube but then the pattern that is imprinted on the capsule has got this funny structure to it and what you're seeing is that that's those are shadows formed by x-rays so the fill tube is sitting here and when the laser beams first enter this cavity the x-ray spots are localized and the fill tube then casts an x-ray shadow onto the capsule and that's what creates this rather large perturbation from a very small fill tube and then it grows and becomes a source of shell material mixed down into the hot spot and I would also point out that in retrospect of course these things are obvious both the effects of shadowing by the fill tube and if I can back up the effects of the tent creating the scar in after the fact after we've seen an experimental piece of data that shows it suddenly it becomes obvious but neither of these two effects as large as they are in the capsule implosions neither were predicted they were quickly understood after a piece of data came in because I was standing there in the hallways when the experimentalist came in and said we look at this this is what I think is happening so it's a nice demonstration of even in the complicated setting like a large laser the experiments still have a very serious role to play and the same thing with this perturbation caused by the fill tube with the spoke patterns coming out they're caused by shadowing and it was only with the experiments then it's obvious you're thinking oh I see that we should have thought of that which we should have but we didn't okay so what we do to try and quantify these effects this is a capsule that has a pre-imposed perturbation machined into it so as it implodes you have a single mold you can study versus just imploding a capsule without any pre-imposed perturbation and you can see depending on whether the capsule was rough or smooth you get more or less growth and also these are the the perturbations caused by those tents and we also try and adjust the x-ray drive to be more forgiving so these initial imperfections don't grow so large so that's the basis of our research and then how we do it there's one there's a technique that's very nice and what it does is that we put some dope gas into the capsule and so when the capsule starts to implode and the shock reaches the center the source of x-rays that we use for radiography are coming from the center of the implosion and so the center of this implosion has a hot spot that radiates and that becomes the source of the x-rays that are going through the capsule and allows us to measure modulations versus radius and so that's called self-backlining with the hot spot x-rays and so this is the type of perturbation we can measure it allows us to measure perturbation growth at the inner surface the inner shell gas interface which is otherwise hard to access and so the way we do it is we use the x-ray source from the hot spot gas itself and this shows a technique where you can imagine in a large program when we were carrying out these measurements what finally gets sorted out is very obvious after the fact it's always not so clear when you're doing the experiments and so for several years there was a big debate amongst all of us the theory, the simulation people and the experimentalists whether or not we were getting capsule shell material mixed down into the hot spot we had a national location campaign we had milestones to meet and so there was a lot of tension about whether to stop and do more careful science experiments or just keep doing performance experiments and so one of the experiments we did that showed conclusively and beyond any argument that there was shell material being mixed down into the hot spot which is supposed to be just D and T but in fact we were getting CH and germanium is that we put a germanium-dope layer in the plastic shell offset from the hot spot by 70 microns of the DT fuel I think is what we have here and so absent strong mixing you would not get germanium down into the hot spot because it was offset by 70 microns albeit prior to compression and so then we did the experiments when the germanium goes down into the hot spot it gets hot and we see germanium spectra x-ray spectra in emission so helium alpha, germanium emission in the hot spot is conclusive proof that shell material that was offset got down in deep enough to get hot and radiate so at that point five years ago or maybe six years ago we said we have a mixed problem we actually are getting shell deep down in the hot spot radiating the heat away and dudding or decreasing the performance so from a science point of view it's beautiful we get spectra, we can analyze it to back out how much material got down into the hot spot and how hot did it get we can also back out what was the effects of the mixing so if you plot up neutron yield for the front against with the sit plotted versus silicon mix carbon plastic carbon silicon mix mass you see that is when the mix mass gets up over up around several hundred nanometers the nuclear yield just drops and plummets and so we call that the mixed cliff and this was a big step forward when we realized in fact we have a mixed problem it's killing us and so we have to slow down and do careful science experiments to quantify it, see how big it is and more importantly, or most importantly learn how to control it in a way that still allows us to get the ignition criteria met more experiments on this so let's see we also can measure the amount of atomic mix in this case we put an inner layer of CD instead of CH just replace the the H with deuterium put it right at the inner layer but then we fill the hot spot gas with tritium so the fill is tritium gas the innermost layer is CD if the D gets down into the T it gets hot then you have DT fusion and you'll get 14 MAV neutrons coming out so this was a nuclear diagnostic and of course indeed we did get DT yield and we also measured that as a function of how far back we offset that deuterated layer so this is the DT nuclear yield that's the mixed signal as a function of the distance that CD layers recessed back away from the hot spot and of course you see the farther back the CD layer is the less of DT it is it's a way to map out very carefully what the actual mix width is because once you've moved far enough away that the mixing stops you know how much distance there was at that direct measure of the mix width or mix extent in such an implosion and so this is getting to the end so if we plot up so here's the plot of perturbation growth as a function of mode number in a capsule implosion where the modes are machined on single mode onto the capsule and imploded convergences of few and what we plot up is the ripple amplitude versus mode number we know what the initial amplitude is so you can turn that into growth factor and this allows you to back up what we would call a dispersion curve where now we can test out various kinds of drives so one kind of drive is very prone to hydrodynamic instability growth that's called what we call it the low foot the leading shock strength was recently small whereas if the leading shock in the drive is much larger it imparts more heat early on it makes the capsule less compressive and it reduces the level of instability growth and so by raising the strength of the leading shock you could lower the amount of rarity of growth but you also lower the compression and so this is the balance we have if we want high compression you risk having high hydrodynamic instability growth and mix if you want to turn down the instability growth and mix you can do so by increasing the adiabat but then the compression drops and of course we need both high compression and high temperature and so that's the sort of the balancing game we have and we have drives that go in between these they're called adiabat shaped that's just playing with the strength of the leading shock and so by varying those leading shock strength we can try and put ourselves in a parameter space more favorable for getting out the nuclear yield that is our goal that's a plot of all that taken together so the one point I would like to raise before I finish is that we have what's called discovery science program and if we allocate about 10% of the laser time to outside users it's an annual call for proposals and it's an issue, it's a call that's issued to the world it's an international call and so if you are interested in or want to think about the possibility of NIF experiments then you should talk to a supremeeer while we're here we can explain how it works what you have to start what kinds of things you need to do to prepare but this is the 10% of the facilities allocated separately for just basic science both internally and outside I mean our inside people compete as well so it's just open, there's no criteria that way and so just talk with us if you have any interest in being involved with your own NIF experiment either anybody's an experiment but in particular you could propose your own okay so these are some of the experiments that are done through that discovery science program we usually call every year so every year there's another team of eight to 10 people that we try and start so this is the latest round and you can see that the types of science that people propose most of these are outside people nuclear science proton beam generation by the TNSA process studying materials at high compression in this case this is carbon as it relates to white dwarfs planar hydromic instabilities, planar direct drive particle acceleration, collision of shock formation which I think Hayesu touched upon in just a bit the properties of brown dwarfs put charged particle stopping powers the equations of state of matter at gigabit pressures and studying the properties of dense hydrogen as you force it to turn liquid or even possibly freeze and so in this 10% that we issue an open call for the types of science really cover a wide space because we basically ask everybody outside if you had access to NIF what would you like to do it's not programmatically driven that's why the types of things we're doing span quite a range okay I think at that point I'm done so I talked about capsule implosions both Hayesu and I have quite an interest in laboratory astrophysics and there's some of this going on through the discovery science program and of course there's a very strong equation of state team as well the studies planetary interiors and planetary formation dynamics as well as the programmatic ICF capsule implosions and so if you're interested in any of this we're here all week and we'd be happy to talk with you thank you


You know, Peter, many country Queenslanders have become great Queensland police officers. And Ian, many of our great footballers have actually started their careers in the Queensland Police Service. Diversity is one of the foundations of our organisation. We provide the opportunity for members of remote and rural communities to join the Queensland Police Service and serve our communities right across Queensland. So why don't you join us at Winton on the 27th and 28th of May at the Queensland Rugby League's Arthur Beats and Outback Shield and learn more about these exciting opportunities. Looking for a rewarding career that I can tell you that the Queensland Police Service has a lot to offer. The Queensland Police Service, your police service, many backgrounds, many pathways, many opportunities and serving Queenslanders everywhere.


Welcome back to the second week of quantum chemistry of atoms and molecules. In the next couple of modules we are going to discuss a rather simple model called particle in a box which helps us understand several nuances of quantum mechanics. Not only that this model also helps us understand the color of carrots. It also helps us understand why is it that the color of semiconductor quantum dots really really small particles nanocrystals of semiconductors why they change color upon changing size. But that all in good time. Let us just recapitulate what we have learned so far. We have talked about Schrodinger wave equation and we got this strange enigmatic parameter called wave function from Schrodinger wave equation. And even though the wave function makes perfect sense in the classical waves it was only after Born provided his interpretation sorry about the error in spelling of interpretation in this slide only after Born provided his interpretation that one can start making sense of the wave function. But Born said if you remember is that amplitude in a regular wave if you take a regular wave and take its amplitude and take mod square of it then you get intensity. So for matter wave also mod psi square should give you a measure of intensity. And then Born said that intensity of matter wave is nothing but the probability density. So probability of finding a particle in one dimensional space let us say between x and x plus dx is given by psi psi star dx. And this probabilistic interpretation of wave function led to several restrictions on wave functions as we have studied already. We know by now that of course psi is a solution of Schrodinger equation but it has to be normalizable. What is the meaning of normalizable? Integrate psi psi star theta over all space that integral has to be equal to 1 because the total probability of finding the particle somewhere in space has to be 1 it has to be somewhere otherwise what are we talking about? Psi must be finite because if it is infinite at even one point then the integral of psi psi star theta is not going to be 1. And also it must vanish at the boundaries this is something that we are going to make use of in the discussions hence forth psi must vanish at the boundaries whatever the boundaries of the system are or at least psi should vanish at plus minus infinity. Then only it is a good solution acceptable solution of Schrodinger equation. Then psi must be continuous we are going to make use of this as well in the present module. It must be a continuous function of spatial coordinates that might be involved because if there is a discontinuity then probability density and therefore probability of the particle at that particular point is undefined this is not acceptable. The first derivative I would not say must be continuous in Q we are going to discuss an example today for particle in a box itself where the first derivative is actually not continuous but psi must be single valued. If for a given value of x psi has 3, 4 different values then it means that probability of finding the particle in that place is well 3, 4 what does that mean that does not make sense. So psi has to be single valued this is something that we will make use of when we talk about particle in a ring and hydrogen atom and psi must be quadratically integrable otherwise you cannot really have it as a solution of Schrodinger equation. And today we are going to establish among other things that all these restrictions that are imposed on the wave functions by on bond interpretation these restrictions lead to quantization. If you remember so far we have not really got quantization in Schrodinger's method. We only have a classical wave equation for de Broglie waves there is no quantum number there whatever wave function that we have discussed so far has no quantum number. So where do the quantum numbers come from? And remember one of the objections against Bohr theory is that the quantum numbers fell from the sky there was no logical explanation it is just that Bohr invoked quantum numbers because Bohr invoked quantum numbers because that was the only way of explaining experimental results. As we will see in Schrodinger's treatment quantum numbers arise naturally the moment we incorporate bond interpretation. We have actually well we said this that we are going to start with talk about exactly solvable systems systems in which Schrodinger equation can be solved exactly without using any approximation. We have started our discussion of free particle a free particle means something that moves without any interaction with anything else. So in this equation of Schrodinger time independent Schrodinger equation we have to set Vx to be equal to 0 because it is a free particle it does not interact with anything else and potential energy if it is there has to arise from interaction with something else. So potential energy is 0 no external force so this is what Schrodinger equation has reduced to as we have discussed in the last module and then we actually went ahead and solved it. It proposed a trial solution of a sin kx plus b cos kx and we verified that the trial solution actually is an eigenfunction of Schrodinger equation.  So plug it back here multiply by minus h cross by 2 m you get an expression of energy to be minus h cross square by 2 m multiplied by minus k square. So this is what we get plus h cross square k square divided by 2 m that is the energy value that we have got and k has turned out to be plus minus root over 2 m e by h cross. We did not say anything about this k we are going to talk about a little more it turns out that this k gives you what the momentum is and as you can see for momentum there is a plus minus sign that means the particle can move in this direction or in this direction in both the directions as shown in this figure. So energy is defined completely magnitude of momentum is defined completely direction of momentum can be either this or that plus or minus we have not really talked about the momentum operator so far we will do it once we get into particle in a box. So this is what it is now there is no restriction on k which means there is no quantization and all energies are allowed. Now let us ask the question is it really a good wave function does it satisfy all the conditions imposed by a bond interpretation and right away I hope you can see that the answer is no. Yes it is continuous no problem with that but does it vanish at the boundaries where is the boundary boundary in this case might be at plus infinity and minus infinity does it vanish does I vanish it does not can you normalize it to be very honest you cannot because limits would be minus infinity to plus infinity that is not going to be one but what one does to circumvent this problem is that for a free particle this wave function is what is called box normalized you have to perform box normalization what does that mean take this wave function let it go to some distance large distance and then you say that this is minus infinity that is plus infinity and this may not be such a bad approximation after all because we are talking about molecular systems atomic systems if you go to even 10 angstroms there is no interaction between 2 atomic particles atomic level particles so infinity does not mean that you have to travel a few light years so that way it is fine but still very strictly speaking it is not such a great wave function but that is what we have got to start with anyway. So with that let us move on a little bit and let us see what happens if you restrict the freedom of the particle let us say we restrict it in a box of a finite length and that takes us to this model of particle in a box that we promised to discuss today what is the model the model is that as long as the particle is between say x equal to 0 and x equal to L the potential energy is 0 which means it is a free particle the moment you go beyond x equal to 0 or beyond x equal to L potential energy is infinity which means that you can think that if you plot potential energy there will be an infinitely high wall at x equal to 0 infinitely high wall at x equal to L so it is as if some selfish giant has entered this quantum mechanical system and built really really tall walls that nobody can climb infinite potential barrier is what we have here and this model right now we are considering only one dimension x so this is called particle in a 1D box it is got a box because it looks like a box and the other name that one can use is a quantum mechanical well with infinitely high potential barrier infinitely high walls all right so let us see what happens to Schrodinger equation for different values of x this is Schrodinger equation that is where we start from for x less than 0 and x greater than L we said that V equal to infinity so we have to substitute infinity for V of x let us do that this is what the equation becomes so if this is the equation then you will get something like infinity multiplied by psi x nothing else matters and then this equation makes sense only when psi of x is equal to 0 right so in other words what we are saying is that the particle cannot exist outside the box and even physically thinking it makes sense that if there is a infinitely high potential energy surface how is it possible potential energy wall how will the particle go outside it cannot as we will see later on if the height is not infinite infinite but finite then the particle can tunnel through it does not have to climb the wall and it tunneling becomes easier if the wall is shorter but let that be the story for another day let us now think what happens when the particle is inside the box x has values between 0 and L for this interval the particle behaves like a free particle so as long as it is inside the box it is free but it cannot go outside the box that is all that is sort of restricted mobility so now see free particle means as you know already V equal to 0 let us put V equal to 0 in the equation we get our own we get back our familiar equation for the free particle just remember now the only difference with the real free particle equation is that it is valid only for x values between 0 and L 0 less than equal to x x is less than equal to L with this interval V is equal to 0 and when V equal to 0 we already know the solution of Schrodinger equation and this is what it is a sin kx plus b cos kx we are not normalized yet but let us start from here a sin kx plus b cos kx and energy we know is h cross k square by 2 m do you have quantization yet not yet but we are not very far away from there now we know what happens outside the box we know what happens inside the box let us worry about the boundary what happens at the wall at x equal to 0 and x equal to L that is what we want to now consider now see outside the box at x less than 0 x greater than L the wave function is 0 and we know one of the conditions imposed by your bond approximation bond interpretation is that the wave function must be continuous so since you go a little if you go a little bit distance beyond x x equal to 0 beyond x equal to L since the wave function is 0 the condition of continuity requires that at x equal to 0 and at x equal to L the wave function has to be 0 right so we will take this x equal to 0 situation first for x equal to 0 psi of x equal to 0 this is required by the boundary condition these are called boundary conditions the boundary condition that the wave function must be continuous all now what happens to this trial solution if your if we use this boundary condition what is the value of psi at x equal to 0 it has to be 0 but then you see when I put x equal to 0 what is sin kx sin kx is also 0 so the first term vanishes no problem with that however the what about the second term in the second term we have b cos kx if you put x equal to 0 then cos 0 is equal to 1 so you are left with psi of x is equal to b in which condition when x equal to 0 so we can say psi at x equal to 0 is equal to b but then we know already from the boundary condition that psi at x equal to 0 has to be 0 so sin psi at x equal to 0 is b it is not difficult to see that b has a value of 0 remember b is a constant a is a constant they are not dependent on the value of x so when b equal to 0 it is 0 for all values of x it is independent of x anyway right that is sort of nice because now the wave function that we have got by using this boundary condition no longer has two terms the second term is eliminated since cos kx is multiplied by b we are left with only one term psi of x is equal to a sin kx okay wave function is simplified do we have quantization not yet but as we said earlier we are not very far away from it right let us now try to use the second boundary condition well I mean the nature of the boundary condition is the same continuity is what we are looking for but we are looking at the other end of the box we have got rid of the cos term by looking at x equal to 0 now what happens if you look at x equal to l even at x equal to l psi has to be equal to 0 same boundary condition continuity so if you put x equal to l then what is the value of psi at l that has to be equal to a sin k l which means since psi at x equal to l equal to 0 a sin k l has to be equal to 0 when sin k l equal to 0 means what what are the possible values k l then must be equal to n pi where n is equal to 1 2 3 4 and so on and so forth integers right so we can now modify this equation and we can write that psi of x is a sin n pi by n pi x by l where n is equal to 1 2 3 4 so on and so forth now for students on the other side of the screen unfortunately I cannot see you but I hope that you have a question at this point of time and the question that you ought to have at this point of time is that what happened to 0 I mean when x equal to 0 then also psi of x is equal to 0 there is no problem in that however mathematics is the tool by which we want to understand the subject it is not the be all and end all so we have to use mathematics and we also have to think what we are dealing with and we are dealing with wave functions we are dealing with wave functions that must satisfy bond interpretation so what happens if you put x equal to 0 sorry n equal to 0 here if you put n equal to 0 then irrespective of the value of x psi becomes 0 right that means within the box psi is 0 everywhere which means mod psi square well there is no mod here it is a real wave function psi square dx is equal to 0 which means probability of finding the particle even inside the box is 0 there is no particle what are we talking about this perhaps the fourth time I have said what are we talking about in this course but well that is what I feel what are we talking about if psi is equal to 0 everywhere that does not make sense that is why n equal to 0 is not what we accept even though it is a valid mathematical solution it is eliminated by bond interpretation so n equal to 1, 2, 3, 4, 5 does it ring a bell what have we achieved we have achieved quantization we have said that only certain wave functions are allowed those wave functions in which n is a positive integer all other wave functions are not allowed as we will see this will lead to quantized energies as well because you might remember that every psi is associated with a particular energy state we will come to that before that let us get done with psi. Let us first determine what the value of a is and let us see what these wave functions would look like for different values of n right to get a value of a what we need to do is normalization when we normalize and now see we had said the generic definition is for normalization the integral has to be from minus infinity to plus infinity but then we know already that from minus infinity to x equal to 0 psi equal to 0 anyway from x equal to l to plus infinity psi equal to 0 anyway so we do not really have to go from minus infinity to plus infinity it is enough if you integrate between the limits 0 to l so integrate between 0 to l psi star of x multiplied by psi of x dx we get a square integral 0 to l sin square n pi by l x dx equal to 1. Now usually students are better than me at mathematics I am sure all of you are and it is not very difficult for you to integrate it but I still do it in steps what we do is we know very well what is the relationship between sin square theta and cos 2 theta we use that relationship and instead of sin square n pi by l x we write 1 minus cos 2 cos 2 n pi by l into x multiplied by half and then we work out this integral what will the integral be I have two terms now right the first one is integral 0 to l dx that is very simple that gives me l and if you take this half it gives me l by 2 the first term becomes l by 2 what about the second term integral of cos kx will give me sin kx multiplied by some constant. Now sin kx integrated from 0 to l what do I get what is the value of sin kx at x equal to 0 it is 0 what is it at x equal to l 0 does remember the integral has the same form as the wave function here so and the wave function vanishes at x equal to 0 and x equal to l so the second term turns out to be 0 so right hand side becomes l by 2 minus 0 now it is very simple we just change the subject of formula a turns out to be root over 2 by l and when we substitute it in the expression for the wave function we get the normalization constant as we said is root 2 by l psi of x turns out to be root 2 by l sin n pi x divided by l so we have got the wave function if I plot the wave functions what do I get these are the wave functions for n equal to 1 what will happen x equal to 0 only at x sorry psi equal to 0 only at x equal to 0 and x equal to l this diagram is a little bit of sync with the labeling here sorry about that and there will be no node node means a point at which the wave function changes sign if I take n equal to 1 n equal to 1 then what will happen the wave function will become 0 not only at the ends not only at the ends but also at l by 2 sorry what am I saying when n equal to 1 there is no node this is the wave function that you get when n equal to 2 then the wave function is root 2 by l sin 2 n pi x divided by l so that function will become 0 even at l by 2 so you will get this you get the full wave earlier we had only half a wave for n equal to 2 we get a full wave for n equal to 3 we get 3 half waves for n equal to 4 we get 2 full waves which means 4 half waves so what we see is that n also stands for the number of half wavelengths that are present for that particular wave function inside the box remember we have said that the wave functions must be orthogonal for a particular system are these wave functions orthogonal well it is very easy to see that if you just plot them let me plot the n equal to 1 well the n equal to 1 wave function is there already let me actually plot here the n equal to wave function on top of the n equal to 2 wave function it will be something like this forgive my poor artistic skills but tell me when I want to search for orthogonality when I want to look for this kind of when I want to look for orthogonality when I want to evaluate this integral what does it actually mean it means that for the 2 wave 2 wave functions have chosen for every point for every value of x I have to multiply this value of psi for psi 1 with psi 2 and I keep on doing that for all and then add up the products that is essentially what I am doing now it is not very difficult to understand that for x equal to 0 to L by 2 product of psi 1 and psi 2 is going to be positive products are going to be all positive for x equal to 0 to sorry for x equal to L by 2 to L these products are all going to be negative so when we add up all the products what do we get we get 0 right so I do not even have to evaluate the integral to say that indeed the wave functions are orthogonal I leave it to you to work this out for any pair of wave functions you desire and then quickly if we want to work it out not by a diagram but actually by evaluating the integral the hard way this is what you do you know we know the relationship between sin C sin D and cos C plus D and cos C minus D we use that relationship work it out and once again since integral of cos is a sin and since the sin functions we know are 0 at x equal to 0 and x equal to L we get a 0 hence it is proved that the wave functions are orthogonal. So what have we got we have got wave functions that are normalized and which form an orthogonal set are they continuous yes they are continuous so far everything looks ankydory let me ask a question now is the first derivative are the first derivatives continuous obviously not because just let us look at this first wave function in equal to 1 what is the first derivative here psi equal to 0 right it is a flat line so first derivative will also be equal to 0 for all these values what is the first derivative here at x equal to L at x equal to L your it is the sin function right psi is still 0 but then go a little bit in what will happen it will be the derivative is going to be cos theta right so if sin theta is equal to 0 what is cos theta 1 sin square theta plus cos theta equal to 1 so suddenly at x equal to L there will be a discontinuity and the value will be 1 of the first derivative so what am I plotting here I am plotting d psi dx against x so and then it will go something like this so there are discontinuities at the boundaries so that is what we had been saying that this continuity of first derivative is not really an essential requirement for the wave function to satisfy born interpretation even though it is put that way in many textbooks okay we stop here today and then we will come back and talk about the energies of these wave functions