How Do You Operate the Hamiltonian on a Coherent State?

tanaygupta2000
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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>
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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!
 
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tanaygupta2000 said:
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...
 
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stevendaryl said:
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.
 
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|
 
Please help!
 
tanaygupta2000 said:
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,...##
 
IMG_20210330_192651.jpg


I tried this way and got struck after this. Please guide.
 
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.
 
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Thank You so much for your help.
 
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tanaygupta2000 said:
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.
 
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