- #1

cyberdeathreaper

- 46

- 0

The question is the following:

Find the eigenfunctions and eigenvalues of a two-dimensional isotropic harmonic oscillator.

Again, I need help simply starting.

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- Thread starter cyberdeathreaper
- Start date

- #1

cyberdeathreaper

- 46

- 0

The question is the following:

Find the eigenfunctions and eigenvalues of a two-dimensional isotropic harmonic oscillator.

Again, I need help simply starting.

- #2

jtbell

Mentor

- 15,946

- 4,616

If you've covered those topics, you should have all the tools you need. Write down the potential energy function for the two-dimensional oscillator, stick it into the two-dimensional Schrödinger equation, and separate the variables to get two one-dimensional equations.

- #3

cyberdeathreaper

- 46

- 0

Just as a general question - once the equation is broken down into two 1D equations, how are the eigenvalues and eigenfunctions obtained? Is it:

[tex]

H \Psi = E \Psi

[/tex]

with Psi being the eigenfunctions and E being the eigenvalues?

- #4

Berislav

- 239

- 0

- #5

cyberdeathreaper

- 46

- 0

Okay, I think I've got it then. Is this correct:

[tex]

\hat{H} = \frac{ (p_x)^2 }{2m} + \frac{ (p_y)^2 }{2m} + \frac{mw^2}{2} \left( x^2 + y^2 \right)

[/tex]

Which is broken up into components:

[tex]

\hat{H} = \hat{H_x} + \hat{H_y}

[/tex]

Noting the 1-D harmonic oscillator gives:

[tex]

E_x = \hbar w \left( n_x + \frac{1}{2} \right)

[/tex]

[tex]

E_y = \hbar w \left( n_y + \frac{1}{2} \right)

[/tex]

[tex]

\psi_n (x) = A_n (a_+)^n \psi_0 (x)

[/tex]

[tex]

\psi_n (y) = A_n (a_+)^n \psi_0 (y)

[/tex]

and noting the separation of variables on the wavefunction:

[tex]

\Psi (x,y) = \Psi_x (x) \Psi_y (y)

[/tex]

Putting this together forms the results - first, using:

[tex]

\hat{H} \Psi = E \Psi

[/tex]

and plugging in the components gives:

[tex]

\left( H_x + H_y \right) \Psi_x \Psi_y = E \Psi_x \Psi_y

[/tex]

So the eigenvalues are:

[tex]

\hbar w \left( n_x + \frac{1}{2} \right) + \hbar w \left( n_y + \frac{1}{2} \right)

[/tex]

Which simplifies to:

[tex]

\hbar w \left( n + 1 \right)

[/tex]

And the eigenfunctions are simply:

[tex]

\psi_n (x) \psi_n (y) = (A_n)^2 (a_+)^{2n} \psi_0 (x) \psi_0 (y)

[/tex]

Is this right - or do I have something wrong here? Thanks for the help.

[tex]

\hat{H} = \frac{ (p_x)^2 }{2m} + \frac{ (p_y)^2 }{2m} + \frac{mw^2}{2} \left( x^2 + y^2 \right)

[/tex]

Which is broken up into components:

[tex]

\hat{H} = \hat{H_x} + \hat{H_y}

[/tex]

Noting the 1-D harmonic oscillator gives:

[tex]

E_x = \hbar w \left( n_x + \frac{1}{2} \right)

[/tex]

[tex]

E_y = \hbar w \left( n_y + \frac{1}{2} \right)

[/tex]

[tex]

\psi_n (x) = A_n (a_+)^n \psi_0 (x)

[/tex]

[tex]

\psi_n (y) = A_n (a_+)^n \psi_0 (y)

[/tex]

and noting the separation of variables on the wavefunction:

[tex]

\Psi (x,y) = \Psi_x (x) \Psi_y (y)

[/tex]

Putting this together forms the results - first, using:

[tex]

\hat{H} \Psi = E \Psi

[/tex]

and plugging in the components gives:

[tex]

\left( H_x + H_y \right) \Psi_x \Psi_y = E \Psi_x \Psi_y

[/tex]

So the eigenvalues are:

[tex]

\hbar w \left( n_x + \frac{1}{2} \right) + \hbar w \left( n_y + \frac{1}{2} \right)

[/tex]

Which simplifies to:

[tex]

\hbar w \left( n + 1 \right)

[/tex]

And the eigenfunctions are simply:

[tex]

\psi_n (x) \psi_n (y) = (A_n)^2 (a_+)^{2n} \psi_0 (x) \psi_0 (y)

[/tex]

Is this right - or do I have something wrong here? Thanks for the help.

Last edited:

- #6

Gokul43201

Staff Emeritus

Science Advisor

Gold Member

- 7,176

- 21

Looks good too (though you haven't said anywhere what n is).

- #7

cyberdeathreaper

- 46

- 0

with n = 0,1,2...

correct?

correct?

- #8

Gokul43201

Staff Emeritus

Science Advisor

Gold Member

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Also [itex]p_x[/itex] should be [itex]\hat{p_x} = \frac {i \hbar}{\sqrt{2m}}\frac {\partial}{\partial x} [/itex]

- #9

qbert

- 185

- 5

So the eigenvalues are:

[tex]

\hbar w \left( n_x + \frac{1}{2} \right) + \hbar w \left( n_x + \frac{1}{2} \right)

[/tex]

Which simplifies to:

[tex]

\hbar w \left( n + 1 \right)

[/tex]

And the eigenfunctions are simply:

[tex]

\psi_n (x) \psi_n (y) = (A_n)^2 (a_+)^{2n} \psi_0 (x) \psi_0 (y)

[/tex]

This is not quite right.

The eigenvalues are right, but there's really no need to simplify

down to n. You have a set of eigenfunctions in x and y with independent

values, and two indices.

[tex] \Psi_{nm} = \psi_n(x) \psi_m(y) = C_{nm} (a_+^n \psi_0(x)) (b_+^m \psi_0(y)) [/tex]

where the "b" operators are exactly analagous to the "a" operators but operate

in y instead of x. So if we want the eigenfunctions that give [itex] p \hbar \omega [/itex]

we need combinations of n and m that add up to p. (That is, there isn't

one eigenfunction for a given "n".)

- #10

cyberdeathreaper

- 46

- 0

I see - so technically the eigenvalues are:

[tex]

\hbar w \left( n + m + 1 \right)

[/tex]

and the eigenfunctions are:

[tex]

\Psi_{nm} = C_{nm} (a_+^n \psi_0(x)) (b_+^m \psi_0(y))

[/tex]

with:

[tex]

C_{nm} = A_n A_m

[/tex]

right?

[tex]

\hbar w \left( n + m + 1 \right)

[/tex]

and the eigenfunctions are:

[tex]

\Psi_{nm} = C_{nm} (a_+^n \psi_0(x)) (b_+^m \psi_0(y))

[/tex]

with:

[tex]

C_{nm} = A_n A_m

[/tex]

right?

Last edited:

- #11

Gokul43201

Staff Emeritus

Science Advisor

Gold Member

- 7,176

- 21

Oops ! Didn't notice that glitch.

cyber : If you are using indexes n,m for the wavefunction, use the same indexes for the eigenvalues. n is your n_x and m is your n_y.

Just also noticed in #5 that an n_y changed into an n_x when writing the eigenvalue.

cyber : If you are using indexes n,m for the wavefunction, use the same indexes for the eigenvalues. n is your n_x and m is your n_y.

Just also noticed in #5 that an n_y changed into an n_x when writing the eigenvalue.

Last edited:

- #12

cyberdeathreaper

- 46

- 0

- #13

Gokul43201

Staff Emeritus

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#10 is now correct.

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