2D LHO calculate ground state energy

[Das apashanka]

A question I have faced in exam to calculate ground state energy
Given Hamiltonian
1/2m(p_x²+p_y²)+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

 

[mfb]

Where does the last term come from?

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

 

[Das apashanka]

Where does the last term come from?

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

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

 

[mfb]

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

 

[Das apashanka]

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

H=1/2m(p_x²+p_y²)+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

 

[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?

 

[Das apashanka]

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?

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

 

[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

 

[mfb]

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

The post was a hint. My first post said basically the same, just a bit more direct. Did you try that?

For the term containing xy if I directly calculate the expectation value using the ground state wave function

You didn't find the ground state wave function yet.

expectation value of x and y is 1/2

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.