Schrödinger equation and interaction Hamiltonian

In summary, the person has been trying to use the Schrödinger equation to reproduce 1A.3 and 1A.4, but has been struggling. They calculated using the assumption that ##\hat{W} = H## but got a different result. They explain their method and ask for clarification on certain aspects. They also provide a reference for the problem and ask for help with a specific statement.
  • #1
TheCanadian
367
13
Screen Shot 2016-05-12 at 11.36.38 AM.png
Screen Shot 2016-05-12 at 11.36.48 AM.png


Given 1A.1 and 1A.2, I have been trying to apply the Schrödinger equation to reproduce 1A.3 and 1A.4 but have been struggling a bit. I was under the assumption that by applying ##\hat{W} \rvert {\psi} \rangle= i\hbar \frac {d}{dt} \rvert{\psi} \rangle## and then taking ##\langle{k'} \lvert \hat{W} \rvert{\psi} \rangle ## and ## \langle{i}\lvert \hat{W} \rvert{\psi} \rangle## would allow me to produce 1A.3 and 1A.4. I may very well be incorrect in my methods, but did the following rough calculation and got a very different result. (In my calculation, I assumed ## \hat{W} = H ##.)

Photo on 2016-05-12 at 12.02 PM.jpg
Any clarification on how to reproduce 1A.3 and 1A.4 would be greatly appreciated.
 
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  • #2
What are ##|i\rangle##, ##|k\rangle##, and ##w##? Moreover, how can you assume that ##H=W##? The answer of all of these questions should be available in the resource you have there.
 
  • #3
blue_leaf77 said:
What are ##|i\rangle##, ##|k\rangle##, and ##w##? Moreover, how can you assume that ##H=W##? The answer of all of these questions should be available in the resource you have there.

## \hat {W} ## is defined as the interaction Hamiltonian. ##|i\rangle## is the initial state the system is prepared in and ##|k\rangle## are all possible states it can evolve into where k is allowed to take all values between ## - \infty ## to ## \infty ## except ##i##.
 
  • #4
My guess is that ##\hat{H} = \hat{H}_0 + \hat{W}##, and that ##\varepsilon## has something to do with the energy in the absence of the interaction (i.e., the eigenvalue of ##\hat{H}_0##.

More details are needed. A reference would be nice.
 
  • #5
DrClaude said:
My guess is that ##\hat{H} = \hat{H}_0 + \hat{W}##, and that ##\varepsilon## has something to do with the energy in the absence of the interaction (i.e., the eigenvalue of ##\hat{H}_0##.

More details are needed. A reference would be nice.

The initial state ## \rvert i \rangle ## has an energy equal to 0, and each state is separated in energy by a difference of ##\varepsilon##. The energy difference between ## \rvert i \rangle ## and ## \rvert k \rangle ## is k##\varepsilon##.

Sure. This is Introduction to Quantum Optics by Grynberg and this problem begins on page 34.
 
  • #6
So ##\hat{H} | k \rangle = k \varepsilon | k \rangle + \hat{W} | k \rangle##. Try that in your method.
 
  • #7
DrClaude said:
So ##\hat{H} | k \rangle = k \varepsilon | k \rangle + \hat{W} | k \rangle##. Try that in your method.

I got it, thank you!

Just one more question, is the statement ##\langle{i} \lvert \hat{W} \rvert{k} \rangle = w_k ## necessarily true if ##\langle{k} \lvert \hat{W} \rvert{i} \rangle = w_k ## ?

If ##\hat{W} ## was an annihilation/creation operator, it seems like this definitely would not be true. Although I had to use both relations written above when reproducing 1A.3 and 1A.4. It might just be something regarding the density matrix that I missed in the text or might be implicitly assumed.
 
  • #8
I haven't done the calculation, but is the idea to calculate ##\left<k'|H|\psi \right>## twice, once using the Schrodinger equation, and once using ##H = H_0 + W## (?) together with the facts that 1) the kets are eigenstates (as in Dr. Claude's post) of ##H_0## and 2) the relations given in the original post.

Edit: TheCanadian posted while I was writing my post. This thread seems to have a surplus of Canadians.
 
  • #9
TheCanadian said:
Just one more question, is the statement ##\langle{i} \lvert \hat{W} \rvert{k} \rangle = w_k ## necessarily true if ##\langle{k} \lvert \hat{W} \rvert{i} \rangle = w_k ## ?

Is ##\hat{W}## Hermitian and

$$w_k = \frac{w}{\sqrt{1+\left(\frac{k \epsilon}{\Delta}\right)^2}}$$

real?
 

What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. It is a partial differential equation that relates the wavefunction of a system to its energy and potential.

What is the interaction Hamiltonian?

The interaction Hamiltonian is a mathematical operator that represents the potential energy of a system due to interactions between its components. It is used in the Schrödinger equation to describe the effects of interactions on the time evolution of a quantum system.

How is the Schrödinger equation used in quantum mechanics?

The Schrödinger equation is used to calculate the wavefunction of a quantum system at any given time, which can then be used to determine properties of the system such as energy, momentum, and position. It is a fundamental tool for understanding and predicting the behavior of quantum systems.

What is the significance of the Schrödinger equation in modern physics?

The Schrödinger equation revolutionized the field of modern physics by providing a mathematical framework for understanding the behavior of quantum systems. It has been used to successfully describe the behavior of particles at the atomic and subatomic level, and has led to many important discoveries and technological advancements.

Can the Schrödinger equation be applied to all quantum systems?

Yes, the Schrödinger equation can be applied to all quantum systems, regardless of their size or complexity. It is a universal equation that accurately describes the behavior of all quantum particles, from individual atoms to large molecules and beyond.

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