Perturbation Theory (Continued)

Homework Continued

Consider the time dependent system from the tutorial represented below (the sections in the figure refer to the sections of the tutorial during which you consider that potential). Let the energy eigenfunctions and eigenvalues of the unperturbed system be given by $φ_m^0 (x)$ and $E_m^0$, respectively.

b) Suppose instead that the system is initially in the state given by $ψ(x,0)=1/5 [4φ_1^0 (x)+3φ_2^0 (x)]$

i) What are the possible results of an energy measurement of this system a $t =T/2$ ? Explain how you would find the probability of each result. Indicate whether or not your probabilities should depend on time.

$ψ(x,0)=1/5 [4φ_1^0+3φ_2^0]$

At $t=T/2$

$ψ(x,t)=c_1 (t) φ_1^0 (x)*exp⁡(-(i E_1^0)/ℏt)+c_2 (t) φ_2^0 (x)*exp⁡(-(i E_2^0)/ℏt)$

So that the possible energies are $E_1^0$  and $E_2^0$.

The probability of each energy is

$P(E_1 )=|c_1 (t) |^2$ and $P(E_2 )=|c_2 (t) |^2$

$(dc_1 (t))/d t=-i/ℏ {c_1*H_{1,1}’+H_{1,2}’*exp⁡(-i*(E_2-E_1)/ℏt)*c_1 (t)}$

$(dc_2 (t))/d t=-i/ℏ {c_2*H_{2,2}’+H_1,2’*exp⁡(i*(E_2-E_1)/ℏt)*c_2 (t)}$

Since the time derivatives are nonzero ($H_{(n,m)}’=<φ_n |V’ | φ_m>$ are nonzero) it means the coefficients will vary with time and also the corresponding probabilities.

ii) What are the possible results of an energy measurement of this system at $t=3T/2$? Explain how you would calculate the probability of each result. Indicate whether or not your probabilities should depend on time.

For $t=3T/2$ the perturbation is turned off, so the discussion is the same as the one from part a) above. All $H_{(n,m)}’$ are zero and the coefficients stay the same as they were at $t=T$. The possible energies are again $E_1^0$  and $E_2^0$ just now the corresponding probability stays constant with time.

iii) What are the possible results of an energy measurement of this system as t approaches infinity? Explain how you would calculate the probability of each result.

iv) Does the probability density for the system change with time during the interval from $t=0$ to $T$ ? Does it change with time when $t>T$ ? Explain.Same considerations as for part a) iii) and iv) above apply.

c) Consider the discussion between three students below.

Student 1: “In both cases, the wave function after the perturbation turns off should look just like it did before the perturbation. If we were in an eigenstate, we should be in the same eigenstate; if we were in a superposition we should be in the same superposition.”

Student 2: ”I disagree. If it started in an eigenstate it will stay there, but if we start in a superposition then the perturbation will cause it to drop into an eigenstate.”

Student 3: “Yes, and if we wait long enough, we will always end up in the ground state no matter what we started with, because the particle will decay.”

With which student do you agree, if any? For each student with whom you disagree, explain the flaws in their reasoning.

Student 1 is wrong. If initially in a mix of states (before perturbation) the mix of states will continue to exist after the perturbation is applied but the probabilities of different states will change with time. Even if the perturbation is turned off the mix of states will continue to stay changed) from the initial value it had before perturbation.

If the mix of states contains initially only one state (an eigenvalue), the applied time dependent perturbation will change the wave function in a true mix of states that will stay the same after turning off the perturbation. In all cases the initial state is not preserved after the perturbation is turned off.

Student 2 is wrong. As discussed above, the perturbation changes the “weight” of the different energies in the mix of states. Each energy “weight” is time dependent as long as the perturbation applied continue to exist.

Student 3 is wrong. If we wait long enough, as long as the perturbation is turned off, the state of the system will remain the same, so no decay into the ground state is possible.

Reference

The Time Dependent Perturbation Theory