Time Dependent Perturbation Theory, Fermi's Golden Rules and Selection Rules - Quantum Mechanics | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Motivation

1. Quantum measurement is a perturbation to quantum system.
2. The perturbation is time dependent.
3. The response of the system induced by perturbation is what we want to calculate. 

Problem

• Stationary solution of unperturbed part

• Time dependent Schrodinger Equation:

 Solution

• The wavefunction can be expanded by unperturbed solutions:

•  Solution of time dependent perturbation:

Two level system

• For a two level system, i.e.  n=1,2

• Solution of time dependent perturbation for two level system:

Perturbation solution  

• Probability of particle at n=2 at time t:

Transition probability 

• Assume initial state is n=1, and probability of transition from n=1 state to n=2 state is:

General derivation ( Interaction Picture)

• We can use so called interaction picture

if there is no perturbation, we have:

namely, in the interaction picture, the state does not change as time if there is no perturbation. Now it is easy to show that

The time dependence of states in the interaction picture depends on H_I .

General derivation ( Interaction Picture) 

• Solution of

We can obtain the solution by iterating. We replace   in the right side of equation according to equation, 

The process can repeat again and again to obtain infinite order.
The first order is given in the first line. Assuming the system at |n> at t=0, the probability for a system at state |m> at time t  is given

Fermi Golden Rule  

• Let’s look the probability in the first order perturbation:

We can show: 

Therefore, we can view g(x,t) as t goes to infinity, 

By taking t to infinite limit, we obtain Fermi Golden Rule Formula:

Transition rate: probability for transition in a unit time interval! 

More about Fermi golden rule

It is an energy conserved process!!
In many cases,  we consider a transition rate associated with an initial state, 

Life time of an state  

• Under the perturbation: the number of particles transfer out of its’ original state are given by:

• Spontaneous emission: normally, a system in excited state is not stable.

The excited state can radiate through interaction with matter. We can introduce a lifetime of an excited state due to spontaneous emission:

•  If both are present:  we have

Selection rule  

• In our formula, the transition between two states is proportional t

If  it is zero, then there is no transition between these two states, namely, The transition between two states is forbidden for this perturbation.

When we know the quantum number or symmetry of the states and know the  symmetry of perturbation potential, then  we can determine the allowed transition. This rule is called selection rule. 

Example one: Scattering in One-Dimension  

• consider a potential barrier in one dimension: 

• For E>V₀, the reflection is given by:,

• We can think the reflection process is from state |k> to |-k> with a time dependent perturbation at zero frequency. Therefore, we can apply the Fermi golden rule:

Example Two: Electromagnetic wave  

• consider a potential barrier in one dimension:

• For given an initial state |i> and final state |f>,  the absorption is 

• The above solution is for monochromatic electromagnetic wave. In general, a wave has a distribution of energy according to frequency,

• Transition rate: 

Example Two: selection in atom

• consider a potential barrier in one dimension: 

• Selection  rule for electromagnetic wave absorption in atom: (matrix element is nonzero)

• For given an initial state |i>=|nlm> and final state |f>=|n’l’m’>, the matrix element

• Proof:  using     operators.