Introduction of Operator and Commutation relation

Operator

An operator is a mathematical symbol representing a certain mathematical rules.

For example:

(1) Parity Operator (\(\hat \pi\)) reflects or inverts the sign of space co-ordinate of wavefunction. i.e.

$$\hat \pi \psi(r)=\psi(-r)$$

$$\Rightarrow [\hat \pi \psi(x,y,z)= \psi(-x,-y,-z)]$$

(2) Identity operator (\(\hat I\)): The operator which leaves the wavefunction unchanged and its eigen value. i.e.

$$[\hat I \psi(r)=1\psi(r)]$$

(3) Null operator (\(\hat O\)): An operator which makes the wavefunction zero or it has eigen value is called null operator. i.e.

$$\hat O\psi(r)=0\psi(r)$$

$$[\hat O\psi(r)=0]$$

In quantum mechanics all the observables quantities corresponds to operators.

For example: Energy operator

$$E_{op}= i\hbar \frac{\partial}{\partial t}\; [Time\;Dependent]$$

$$\hat H=\frac{-\hbar^2}{2m}\nabla^2+\hat V(r)\; [ Time\; independent ]$$

Position Operator (\(\hat r\))= r

Linear momentum operator (\(\hat P_x)=-i\hbar\nabla\) etc.

Commutation Relation:

Commutator

If \(\hat A\) and \(\hat B\) are any two operators then the expression \(\hat A\hat B- \hat B\hat A\) is known as commutator of \(\hat A \) and \(\hat B\). It is also an operator. It is represented as

$$\hat A\hat B-\hat B \hat A=[\hat A, \hat B]\dotsm(1)$$

Properties of Commutation Relation:-

(1) For any two operator \(\hat A\) and \(hat B\):-

$$[\hat A, \hat B]= -[\hat B, \hat A]$$

Proof:- $$[\hat A, \hat B]= [\hat A\hat B- \hat B\hat A]$$

$$=-[\hat B\hat A- \hat A\hat B]$$

$$=-[\hat B, \hat A]$$

$$Proved$$

(2) For any three operators:-

$$[\hat A, \hat B+ \hat C]=[\hat A, \hat B]+ [\hat A, \hat c]$$

Proof:-

$$[\hat A, \hat B+ \hat C]=\hat A(\hat B+ \hat C)-(\hat B+ \hat c)\hat A$$

$$=\hat A\hat B+\hat A\hat C-\hat B\hat A-\hat C\hat A$$

$$=(\hat A \hat B- \hat B\hat A)+ (\hat A\hat C- \hat C\hat A)$$

$$=[\hat A,\hat B]+[\hat A, \hat C]$$

$$=Proved$$

(3) $$[\hat A+ \hat B, \hat C]= [\hat A, \hat C]+ [\hat B, \hat C]$$

Proof:

$$[\hat A+ \hat B, \hat C]= (\hat A+ \hat B) \hat C- \hat C(\hat A+ \hat B)$$

$$=\hat A\hat C+ \hat B\hat C- \hat C\hat A- \hat C\hat B$$

$$=\hat A\hat C- \hat C\hat A+ \hat B\hat C-\hat C\hat B$$

$$=[\hat A,\hat C]+ [\hat B,\hat C]$$

$$=Proved$$

(4) $$[\hat A, \hat B\hat C]= \hat B[\hat A, \hat c]+ [\hat A, \hat B]\hat C$$

Proof:

$$[\hat A, \hat B\hat C]= \hat A(\hat B\hat C)-(\hat B\hat C)\hat A$$

$$=\hat A\hat B\hat C - \hat B\hat C\hat A+ \hat B \hat A\hat C - \hat B\hat A\hat C$$

Adding and subtracting \(\hat B\hat A\hat C\)

$$=(\hat B\hat A\hat C- \hat B\hat C \hat A)+ \hat A \hat B\hat C- \hat B\hat A \hat C$$

$$=\hat B(\hat A \hat C- \hat C\hat A)+ ( \hat A \hat B- \hat B\hat A) \hat C$$

$$=\hat B[\hat A, \hat C]+[\hat A, \hat B]\hat C$$

$$Proved$$

(5) $$[\hat A\hat B, \hat C]= \hat A[\hat B,\hat C]+[\hat A,\hat C]\hat B$$

Proof: $$[\hat A\hat B, \hat C]= (\hat A\hat B)\hat C- \hat C(\hat A\hat B)$$

$$=\hat A \hat B\hat C- \hat C\hat A\hat B+ \hat A\hat C\hat B-\hat A \hat C\hat B$$

$$\hat A(\hat B\hat C-\hat C\hat B)+ (\hat A\hat C- \hat C\hat A)\hat B$$

$$=\hat A[\hat B,\hat C]+[\hat A,\hat C]\hat B$$

$$Proved$$

(6)$$[\hat A,\alpha \hat B]= \alpha [ \hat A, \hat B] $$

Where, \(\alpha\) is a constant numbers.

Proof:

$$[\hat A,\alpha \hat B]= \hat A(\alpha \hat B)- (\alpha \hat B) \hat A$$

$$=\alpha \hat A\hat B-\alpha \hat B\hat A$$

$$=\alpha (\hat A\hat B- \hat B\hat A)$$

$$\alpha[\hat A, \hat B]$$

$$Proved$$

(7)For any three operator

$$[\hat A, [\hat B, \hat C]]+ [\hat C, [\hat A, \hat B]]+ [\hat B[\hat C,\hat A]]=0 $$

This is called Jacobian identity.

Proof:

$$[\hat A, [\hat B, \hat c]]+ [\hat C, [\hat A,\hat B]]+ [\hat B, [\hat C, \hat A]]$$

$$[\hat A, \hat B\hat C- \hat C\hat B]+[\hat C, \hat A\hat B -\hat B\hat A]+[\hat B, \hat C\hat A- \hat A\hat C]$$

$$\hat A(\hat B\hat C-\hat C\hat B)- (\hat B\hat C- \hat C\hat B)\hat A+ \hat C(\hat A\hat B- \hat B\hat A)-(\hat A\hat B- \hat B\hat A)\hat C+\hat B(\hat C\hat A- \hat A\hat C)- (\hat C\hat A- \hat A\hat C)\hat B$$

$$=\hat A\hat B\hat C- \hat A\hat C\hat B-\hat B\hat C\hat A+ \hat C\hat B\hat A+ \hat C\hat A\hat B-\hat C\hat B\hat A-\hat A\hat B\hat C+\hat B\hat A\hat C+\hat B\hat C\hat A-\hat B\hat A\hat C-\hat C\hat A\hat B+\hat A\hat C\hat B$$

$$=0$$

$$Proved$$

Physical Meaning of commutator:

If two operators \(\hat A\) and \(\hat B\) are commutate, then they can be measured simultaneously Otherwise they are called canonically conjugate variables and can not be measured simultaneously.

Reference:

1. Mathews, P.M and K Venkatesan. A Text Book of Quantum Mechanics. New Delhi: Tata McGraw Hill Publishing Co. Ltd, 1997.
2. Merzbacher, E. Quantum Mechanics . New York: John Wiley, 1969.
3. Prakash, S and S Salauja. Quantum Mechanics. Kedar Nath Ram Nath Publishing Co, 2002.
4. Singh, S.P, M.K and K Singh. Quantum Mechanics. Chand & Company Ltd., 2002.