**Required math: calculus **

**Required physics: **Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.12.

Shankar, R. (1994), *Principles of Quantum Mechanics*, Plenum Press. Section 7.4, Exercise 7.4.2.

In the study of the harmonic oscillator, we can express and in terms of the raising and lowering operators:

We now have

The reason this is zero is that, as we saw when working out the normalization of the stationary states,

and since the wave functions are orthogonal, we get

Similarly:

for the same reason.

For the mean squares:

In going from the first to the second line, we’ve thrown out terms where we integrate two orthogonal functions. For example,

We have used the relations above and the fact that is normalized to get the third line.

Similarly:

The uncertainty principle then becomes

and the kinetic energy is

which is half the total energy, as it should be.

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LukeWhat happens to the normalization factor 1/sqrt(n!) in phi_n?

LukeNevermind, I half wrote out phi_n as a term of phi_0’s and forgot about it

apashanka dassir my question is,the ground state wave function which is here is actually the representation of the ground state wave function in the basis of eigenvectors of position operator,how can it be total wave function?please explain ,thank you

apashanka dassir N=a^a(the raising and lowering operator) is having 0 eingenvalue for ground state?

apashanka dassir what will be the representation of position and momentum operator in the basis of eigenvectors of position operator?please reply ,thanks

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