Time Dependent Perturbation Theory, Fermi's Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

Physics : Time Dependent Perturbation Theory, Fermi's Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

The document Time Dependent Perturbation Theory, Fermi's Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
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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. 

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Problem

• Stationary solution of unperturbed part

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

• Time dependent Schrodinger Equation:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 Solution

• The wavefunction can be expanded by unperturbed solutions:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  •  Solution of time dependent perturbation:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Two level system

  • For a two level system, i.e.  n=1,2

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

• Solution of time dependent perturbation for two level system:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Perturbation solution  

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Transition probability 

  • Assume initial state is n=1, and probability of transition from n=1 state to n=2 state is:Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

General derivation ( Interaction Picture)

  • We can use so called interaction picture

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

if there is no perturbation, we have:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

 

General derivation ( Interaction Picture) 

  • Solution of

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

We can obtain the solution by iterating. We replace Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev  in the right side of equation according to equation, 

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Fermi Golden Rule  

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

We can show: Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

 

More about Fermi golden rule

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Life time of an state  

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  • 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:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  •  If both are present:  we haveTime Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Selection rule  

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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: Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  • For E>V0, the reflection is given by:,Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  • 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:

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Example Two: Electromagnetic wave  

  • consider a potential barrier in one dimension:Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  • Transition rate: 

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

 

Example Two: selection in atom

  • consider a potential barrier in one dimension: Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  • 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

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev

  • Proof:  using Time Dependent Perturbation Theory, Fermi`s Golden Rules and Selection Rules - Quantum Mechanics Physics Notes | EduRev    operators.

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