Hamiltonian of a displaced QHO

[tanaygupta2000]

Homework Statement

For α = f/(ω√2mћω), evaluate H|α> for H = p^2/2m + mω^2x^2/2 - fx, which is the Hamiltonian of a displaced Harmonic Oscillator under a constant force f. Is this |α> the ground state of this displaced simple harmonic oscillator?

Relevant Equations

H|α> = En|α> = (n + 1/2)ћω |α>

I am getting that we have to operate the given Hamiltonian on the given state |α>. But what is confusing me is that since this H contains position and momentum operators which just involve variable x and partial derivative, how do I operate this H on the given α, since it seems like α is essentially a constant quantity? Please help!

 

[stevendaryl]

Homework Statement:: For α = f/(ω√2mћω), evaluate H|α> for H = p^2/2m + mω^2x^2/2 - fx, which is the Hamiltonian of a displaced Harmonic Oscillator under a constant force f. Is this |α> the ground state of this displaced simple harmonic oscillator?
Relevant Equations:: H|α> = En|α> = (n + 1/2)ћω |α>

View attachment 280440

I am getting that we have to operate the given Hamiltonian on the given state |α>. But what is confusing me is that since this H contains position and momentum operators which just involve variable x and partial derivative, how do I operate this H on the given α, since it seems like α is essentially a constant quantity? Please help!

Have you been through the exercise of defining the "raising and lowering" operators for the harmonic oscillator? If so, then you should know that $x$ and $p$ can be expressed as linear combinations of the raising and lowering operators.

Alternatively, if you solve the harmonic oscillator in position space, then you should know that the state satisfying $H |a\rangle = (n+1/2)\hbar \omega |a \rangle$ can be expressed in the position basis as a function of $x$. That's a lot more complicated way to go, so I'm assuming that you are using the raising and lowering operators...

 

[tanaygupta2000]

Have you been through the exercise of defining the "raising and lowering" operators for the harmonic oscillator? If so, then you should know that $x$ and $p$ can be expressed as linear combinations of the raising and lowering operators.

Alternatively, if you solve the harmonic oscillator in position space, then you should know that the state satisfying $H |a\rangle = (n+1/2)\hbar \omega |a \rangle$ can be expressed in the position basis as a function of $x$. That's a lot more complicated way to go, so I'm assuming that you are using the raising and lowering operators...

Yes, sir! I am very much familiar with the representation of position and momentum operators in terms of raising and lowering operators:
x = √(hbar/2mw) (a' + a)
p = i√(mhw/2) (a' - a)

where x and p are position and momentum operators and a and a' are creation and annihilation operators.

 

[tanaygupta2000]

But I think a and a' only act on |n>.
Or should I use this property for coherent states:
a|α> = α|α>
<α|a' = α*<α|

 

[tanaygupta2000]

Please help!

 

[stevendaryl]

But I think a and a' only act on |n>.
Or should I use this property for coherent states:
a|α> = α|α>
<α|a' = α*<α|

$|a\rangle$ is the state $|n\rangle$. It's not a coherent state. Your "relevant equation" says:

$H |a\rangle = (n+1/2) \hbar \omega |a\rangle$

There is some confusion here, because you're using the same symbol, $H$ to mean both the original hamiltonian and the "displaced" hamiltonian. What I assumed that the problem was asking for was

Let $H_0 = p^2/2m + m \omega^2/2 x^2$.
Let $|n\rangle$ be a state such that $H_0 |n\rangle = (n+1/2) \hbar \omega |n\rangle$.
Let $H = H_0 - fx$.

Then what is $H |n\rangle$ in terms of the original basis, $|n\rangle$?

So under this interpretation of the question, $H|n\rangle =$ some combination of states $|0\rangle, |1\rangle, |2\rangle,...$

 

[tanaygupta2000]

I tried this way and got struck after this. Please guide.

 

[stevendaryl]

I think that that’s as much as you can do. The ground state of the displaced Hamiltonian cannot be equal to $|n\rangle$, since acting on it by the Hamiltonian mixes states of different $|n\rangle$.

You could try to find the ground state of the displaced Hamiltonian, but that’s going beyond what the problem asked for.

 

[tanaygupta2000]

Thank You so much for your help.

 

[vela]

I am getting that we have to operate the given Hamiltonian on the given state |α>. But what is confusing me is that since this H contains position and momentum operators which just involve variable x and partial derivative, how do I operate this H on the given α, since it seems like α is essentially a constant quantity? Please help!

I had the same question when I read the problem statement. What is the relationship between $\alpha$ and $\lvert \alpha \rangle$? Have you asked your instructor for clarification?

I think your interpretation that the question is about coherent states of the harmonic oscillator may be correct. In that case, the state $\lvert \alpha \rangle$ satisfies $\hat a \lvert \alpha \rangle = \alpha \lvert \alpha \rangle$. You can find $E_0$, the energy of the ground state of the displaced harmonic oscillator, by completing the square in the Hamiltonian.