How Does Time-Dependent Perturbation Theory Explain Quantum State Evolution?

In summary, the conversation discusses a system with a small perturbation and its solutions, both with and without the perturbation. It then presents two equations and shows that there may be a mistake in one of the equations, which is eventually corrected. The conversation ends with a question about obtaining solved papers on quantum mechanics.
  • #1
kakarukeys
190
0
Given a system,
[itex]H = H_0 + V[/itex]
V is a small perturbation that does not depend on time.

the system is in [itex]|E_0>[/itex] at time [itex]t_0[/itex]
[itex]H_0 |E_n> = E_n |E_n> [/itex]
[itex]H_0 |E_0> = E_0 |E_0> [/itex]

Let [tex]|\Psi(t)>[/tex] be the solution of the system.
Let [tex]|\Phi(t)>[/tex] be the solution of the system without perturbation.
Let [tex]|u(t)> = |\Psi(t)> - |\Phi(t)>[/tex].

Show that [tex]|<E_n|u(t)>|^2 = 4 |V_{n0}|^2 [{{\sin(t - t_0)(E_n - E_0)/\hbar}\over{E_n - E_0}}]^2[/tex]

at lowest order
No matter how many times I try, the answer I get is

[tex]|<E_n|u(t)>|^2 = 2 |V_{n0}|^2 [{{1 - \cos(t - t_0)(E_n - E_0)/\hbar}\over{E_n - E_0}}]^2[/tex]

Please help!
 
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  • #2
I think you got the right answer, but u don't have the correct question, it should be show that ...sin(.../2), there is a divide by 2 missing in the sine function. I looked it up in my quantum book.
 
  • #3
Sorry,
I typed wrongly,

My answer was
[tex]|<E_n|u(t)>|^2 = 2 |V_{n0}|^2 [{{1 - \cos(t - t_0)(E_n - E_0)/\hbar}\over{(E_n - E_0)^2}}][/tex]

Ya, you are brilliantly right,

Since,
[tex]2\sin^2\theta/2 = 1 - \cos\theta[/tex]

There should be a "divided by 2" inside the Sine
 
  • #4
How can i get certain solved papers in quantum mechanics?
thanks
 

FAQ: How Does Time-Dependent Perturbation Theory Explain Quantum State Evolution?

What is time-dependent perturbation?

Time-dependent perturbation is a mathematical technique used in quantum mechanics to study the effects of a time-varying perturbation on a quantum system. It allows us to calculate the probability of transitions between different energy states of the system over time.

How is time-dependent perturbation different from time-independent perturbation?

Time-dependent perturbation deals with perturbations that vary with time, while time-independent perturbation deals with perturbations that are constant. In time-dependent perturbation, the system is in a superposition of energy states, while in time-independent perturbation, the system is in a particular energy state.

What are some examples of time-dependent perturbations?

Some examples of time-dependent perturbations include oscillating electric or magnetic fields, time-varying potentials, and electromagnetic radiation. These perturbations can affect the energy levels of a quantum system and cause transitions between them.

How is time-dependent perturbation used in real-world applications?

Time-dependent perturbation is used in many areas of physics and chemistry, including spectroscopy, nuclear magnetic resonance, and quantum computing. It is also used in studying the behavior of atoms and molecules in external fields, such as in laser-matter interactions.

What are the limitations of time-dependent perturbation?

One limitation of time-dependent perturbation is that it assumes the perturbation is small and does not significantly change the system. It also does not take into account the effects of quantum fluctuations. Additionally, it can be challenging to calculate the exact analytical solutions for more complex systems, and numerical methods may be required.

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