# Time dependant perturbation

• quasar987
In summary, the conversation discusses the nature of time-dependent perturbation theory and the calculation of probabilities of finding a system in a specific eigenstate. The main point of confusion is that at a certain time, the Hamiltonian is no longer the same as the initial one, causing some eigenstates to no longer be valid. However, the theory still works by considering the interaction to only be switched on for a short period of time, allowing the use of the initial eigenstates as approximations.

#### quasar987

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Gold Member
Hi,

I have a problem with the very nature of time dependant perturbation theory (TDPT). In TDPT, we consider a system of Hamiltonian H(t) = H_0 (for t<0), H(t)=H_0 + kW(t) (for t>0) [where k<<1], where H_0 is, for simplicity, discrete, non-degenerate and time-independent, and given that at t=0, the state of the system is |phi_i> (an eigenstate of H_0), we are interested in calculating P_if(t), the probability of finding the system in another eigenstate of H_0, |phi_f>, at time t.

But this does not make sense because at t, the hamiltonian is no longer H_0, so it will, in general, not have |phi_f> as an eigenstates. But we know that the result of a measurement will project the wave-function into one of the eigenfunction. So as soon as |phi_f> is not an eigenstate of H(t), the probability of finding the system in |phi_f> will be 0.

"So as soon as |phi_f> is not an eigenstate of H(t), the probability of finding the system in |phi_f> will be 0"

I'm not sure if i fully understand your question, but why does it worry you that |phi_f> will not in general be an eigenstate of H(t)? If i remember my last quarter of undergrad quantum correctly we are only worried about the probability of transition between |phi_i> and |phi_f> (both of which are eigenstates of H_0) under a time dependent pertubation that takes place for some delta t. As far as the results of measurements "forcing" the wave-function's projections onto one of the eigenstates, simply think about the TDP as having already measured the system before (hence our eigenstate |phi_i>) as well as after, (hence |phi_f>) and the TDP simply spits out the probablity of this transition within the abovementioned delta t.

Gza said:
"So as soon as |phi_f> is not an eigenstate of H(t), the probability of finding the system in |phi_f> will be 0"

I'm not sure if i fully understand your question, but why does it worry you that |phi_f> will not in general be an eigenstate of H(t)? If i remember my last quarter of undergrad quantum correctly we are only worried about the probability of transition between |phi_i> and |phi_f> (both of which are eigenstates of H_0) under a time dependent pertubation that takes place for some delta t. As far as the results of measurements "forcing" the wave-function's projections onto one of the eigenstates, simply think about the TDP as having already measured the system before (hence our eigenstate |phi_i>) as well as after, (hence |phi_f>) and the TDP simply spits out the probablity of this transition within the abovementioned delta t.

Like you said, we have measured the system before the perturbation to be |phi_i>. Then the TDP is turned on and the wave function evolves according to the time dependant SE

$$i\hbar\partial_t \Psi(t)=H(t)\Psi(t)$$

with initial condition Psi(0)=phi_i. Now say we measure the energy at a time t. The wave function Psi(t) will collapse into an eigenstate of H(t). It is most likely that phi_f will not be such an eigenstate for H(t), so let's suppose for simplicity that it isn't.

Sure, mathematically, nothing stops us from expanding Psi(t) in a Fourier series in |phi_n> and we can even calculate the coefficient |<Psi(t)|phi_k>|², but it does not represent the probability of finding the system in state |phi_k>. According to the expansion postulate, this probability is 0, because phi_k is not an eigenstate of H(t).

But the books say that the probability is |<Psi(t)|phi_k>|², so this is where I'm confused, and I'm asking "where in the above am I mistaken"?

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quasar987 said:
But this does not make sense because at t, the hamiltonian is no longer H_0, so it will, in general, not have |phi_f> as an eigenstates. But we know that the result of a measurement will project the wave-function into one of the eigenfunction. So as soon as |phi_f> is not an eigenstate of H(t), the probability of finding the system in |phi_f> will be 0.

You should actually consider the following:
for t<0, we have H0, and for t>t1, we also have H0 as hamiltonian.
We only consider the interaction "switched on" between t=0 and t=t1.
We prepare the state at t<0 and we measure it after t=t1.

The idea is that H0 is "good enough" as eigenstate generator, but not as time evolution generator. So we consider that, for short enough times, we can take the eigenstates of H0 as good approximations to the eigenstates of the full hamiltonian, but that the neglected parts do play a cumulative role in skewing this as time evolution.

## 1. What is time dependent perturbation theory?

Time dependent perturbation theory is a mathematical framework used to study the behavior of quantum systems that are subjected to a time-dependent external perturbation. It allows us to calculate the changes in the system's energy levels and wavefunctions over time.

## 2. What are the assumptions of time dependent perturbation theory?

The main assumptions of time dependent perturbation theory are that the perturbation is small enough to be treated as a perturbation, and that the perturbation varies smoothly and slowly with respect to time.

## 3. How is time dependent perturbation theory different from time independent perturbation theory?

Time dependent perturbation theory is used for systems that are subjected to a time-dependent external perturbation, while time independent perturbation theory is used for systems with a time-independent perturbation. In time dependent perturbation theory, the system's energy levels and wavefunctions change over time, while in time independent perturbation theory, they do not.

## 4. What is the time evolution operator in time dependent perturbation theory?

The time evolution operator in time dependent perturbation theory is a mathematical operator that describes the time evolution of a quantum system under the influence of a time-dependent external perturbation. It is represented by the unitary operator U(t) and is used to calculate the system's time-dependent wavefunction.

## 5. How is time dependent perturbation theory used in practical applications?

Time dependent perturbation theory is used in many areas of physics, including atomic and molecular physics, condensed matter physics, and quantum chemistry. It is used to study the behavior of systems under the influence of time-dependent external fields, such as electromagnetic fields, and has practical applications in areas such as laser technology and nuclear magnetic resonance imaging.