Question

Harmonic Oscillator in Quantum Mechanics

Answer 1

Ground state \[\Psi_0 = N_0e^{-\alpha x^2/2}\]

Answer 2

First excited state\[\Psi_1 = N_0 x e^{- \alpha x^2/2}\]

Answer 3

Is this for a 1D H.O?

Answer 4

What are the physical dimensions of N0 andN1? How would you calculate these constants> Write the formal expression.

Answer 5

I think its 1D

Answer 6

Keep it simple

Answer 7

normalize the function.

Answer 8

you were given to it as a Gaussian function, which satisfies the requirement of going to zero at infinity, making it possible to normalize the wavefunction.

Answer 9

In class we discussed the equations \[\Psi _0 = [\frac{\alpha }{\pi}]^ {1/4}e^{- \alpha x^2/2}\]

Answer 10

yes. thats the normalized wavefunction if i remember

Answer 11

and \[\Psi_1 = [\frac{[4 \alpha^3]}{\pi}]^{1/4} x e^{- \alpha x^2/2}\]

Answer 12

\(\sf \Psi _0 = [\frac{\alpha }{\pi}]^ {1/4}e^{-\frac{ \alpha x^2}{2}}\) and

\(\sf \Psi _1 = [\frac{\alpha }{\pi}]^ {1/4}\sqrt{2} e^{-\frac{ \alpha x^2}{2}}\)

Answer 13

So, I though well

\[\int\limits_{- \inf}^{\inf}\left| \Psi \right|^2 = 1\]

Answer 14

Well, you can estimate since these are just probabilities. You don't NEED to integrate.

Answer 15

Well N is to normalize the function. So how do I even get the physical dimensions.

Answer 16

the sum of the probability of both these wave functions is all that can exist for the position of the particle

Answer 17

to get a physical meaning, you square it, because the wave function itself has no physical meaning.

Answer 18

^

Answer 19

exactly but...what are the physical units I am squaring.

Answer 20

harmonic oscilattor takes a sinusoidal behavior, I think.

Answer 21

I think @dan815 might be more famliar with this. It's been a looong time since i've done physical chemistry.

Answer 22

I think its based on the classical harmonic oscillator

Answer 23

the position saquare is very similiar to
x=x_bar-deltax
and this show up in statistics remember.. when you tried to calculate SD. or variance which is square of sd!
this is very similar to a variance calculation