How Does Time-Dependent Perturbation Impact Quantum States?

[maria clara]

Homework Statement

V= V₀ (r) + V₁(r,t)

V₀ (r) =-e^2/r

V₁(r,t) is a small perturbation which is being activated only in the interval 0<t<t_f

The system starts in the ground state, where l =0

1. If the change in the potential is very slow, what is the probability of finding the system at
t_f in the state where l = 1?
2. The same question only now consider the case where the change is fast.
3. What is the condition that t_f needs to satisfy in order to be small enough as to be considered as a fast perturbation?

2. The attempt at a solution

1. The change is adiabatic so the state of the system does not change. Thus it remains in the ground state and the probability to find the system in another state is zero.
2. Here the change very quick, so the wave function doesn't change at all, so again the transition probability is zero.
But here I need your help - I get the same transition probability in both cases. What is the difference between the two systems (the one that changed adiabatically and the one that changed rapidly) at t_f? they both remain in an eigenstate of the Hamiltonian operator, but the eigenenergies are different? Does the wavefunction in the first case change?
3. It is clear that t_f should be smaller than some characteristic time of the system, but how do I find it?

Thanks in advance!

 

[olgranpappy]

Homework Statement

V= V₀ (r) + V₁(r,t)

V₀ (r) =-e^2/r

V₁(r,t) is a small perturbation which is being activated only in the interval 0<t<t_f

The system starts in the ground state, where l =0

1. If the change in the potential is very slow, what is the probability of finding the system at
t_f in the state where l = 1?
2. The same question only now consider the case where the change is fast.
3. What is the condition that t_f needs to satisfy in order to be small enough as to be considered as a fast perturbation?

2. The attempt at a solution

1. The change is adiabatic so the state of the system does not change. Thus it remains in the ground state and the probability to find the system in another state is zero.
2. Here the change very quick, so the wave function doesn't change at all, so again the transition probability is zero.
But here I need your help - I get the same transition probability in both cases. What is the difference between the two systems (the one that changed adiabatically and the one that changed rapidly) at t_f? they both remain in an eigenstate of the Hamiltonian operator, but the eigenenergies are different? Does the wavefunction in the first case change?
3. It is clear that t_f should be smaller than some characteristic time of the system, but how do I find it?

Thanks in advance!

I'm not familiar with your notation. What is I?

Is it an energy?

Is there a scale in this problem? What is it? What are it's units? Can you use fundamental constants to turn it into a time scale?

 

[maria clara]

Hi olgranpappy,

I copied the question as it is.
It isn't the Capital letter I, it's a lowercase "L" - l, the second quantum number.

I guess that the "system" is a hydrogen atom, though it isn't explicitly stated in the problem.


thanks in advance.

 

[olgranpappy]

Hi olgranpappy,

I copied the question as it is.
It isn't the Capital letter I, it's a lowercase "L" - l, the second quantum number.

Ah. Okay.

I guess that the "system" is a hydrogen atom, though it isn't explicitly stated in the problem.

The V_0 is the potential seen by an electron in the hydrogen atom. So, it is implicit that we are talking about hydrogen atom orbitals as the unperturbed basis.

"The state with l=1" does not make sense because there are many states with l=1. Presumably the question is asking about transtitions to the lowest state with l=1?

thus the energy scale involved is the difference in energy between the ground state and the first excited state. this can be turned into a time scale by using Planck's constant.

 

[maria clara]

OK, so I guess I should just take hbar/\DeltaE as the characteristic time of the system.
But what about the difference between the two systems described abouve at tf?

Thanks(:

 

[olgranpappy]

OK, so I guess I should just take hbar/\DeltaE as the characteristic time of the system.
But what about the difference between the two systems described abouve at tf?

how have you attempted to answer this question on your own? can you show some work?