If one defines a † † = 1 2 ( − − d d q + q ) as the "creation operator" or the "raising operator" and a = 1 2 ( d d q + q ) as the "annihilation operator" or the "lowering operator", the Schrödinger equation for the oscillator reduces to ℏ ℏ ω ω ( a † † a + 1 2 ) ψ ψ ( q ) = E ψ ψ ( q ) . This is significantly simpler than the original form. Further simplifications of this equation enable one to derive all the properties listed above thus far.