# I 2D LHO

1. Dec 12, 2017

### Das apashanka

A question I have faced in exam to calculate ground state energy
Given Hamiltonian
1/2m(px2+py2)+1/4mw2(5x^2+5y^2+6xy)
ground state energy has to be obtained
Its clear that the Hamiltonian is a 2D LHO Hamiltonian but what for the term 3/4(x+y)2

2. Dec 13, 2017

### Staff: Mentor

Where does the last term come from?

It can be interesting to introduce mew variables for the sum and difference of x and y.

3. Dec 13, 2017

### Das apashanka

I have broken the terms and got hamiltonian of a 2D LHO plus that term

4. Dec 13, 2017

### Staff: Mentor

It would help if you could show the work instead of describing it like that.

5. Dec 13, 2017

### Das apashanka

H=1/2m(px2+py2)+mw2/2(x2+y2)+3/4(x2+y2+2xy)
where first two terms are of 2D LHO ,and there is the last term
The ground state energy is to be calculated

6. Dec 13, 2017

### DrDu

The kinetic term is invariant under an orthogonal transform of the variables x and y. Can you use this to bring the potential term to a more standard form?

7. Dec 13, 2017

### Das apashanka

Mr Dr Du will you please give some kind of hints to solve this problem

8. Dec 13, 2017

### Das apashanka

For the term containing xy if I directly calculate the expectation value using the ground state wave function the answer is coming as 3mw2/8 which actually dimensionally doesn't match
Since <Φ0(x)Φ0(y)I3mw2xy/2IΦ0(x)Φ0(y)>
expectation value of x and y is 1/2​

Is there any discrepancy regarding this

Last edited by a moderator: Dec 13, 2017
9. Dec 13, 2017

### Staff: Mentor

The post was a hint. My first post said basically the same, just a bit more direct. Did you try that?
You didn't find the ground state wave function yet.
The function is symmetric with respect of x and y, the expectation value for both is zero. They are correlated, the expectation value of xy doesn't have to be zero. But you are not yet at the step where you can calculate these things.