Properties of parity operator and Hermitian operator

Conti: Properites of Hermitian operator

4. Two eigen functions of a Hermitian operator have different eigen values are orthogonal.

\(\Rightarrow\)

Let \(\psi_1\) and \(\psi_2\) be two different eigen function of a Hermitian operator \(\hat A\) having eigen values \(\lambda_1\) and \(\lambda_2\) respectively.

The eigen value equation of \(\hat A\) are

$$\hat A\psi_1= \lambda_1 \psi_1\dotsm(1)$$

$$\hat A\psi_2= \lambda_2 \psi_2\dotsm(2)$$

Here, \(\lambda_1- \lambda_2\ne 0\) and we have to shown that

$$\int \psi_1^* \psi_2 dx=0$$

$$OR$$

$$\int \psi_2^* \psi_1 dx=0$$

Taking complex conjugate of equation (1) and (2)

$$[\hat A \psi_1]^*= [\lambda _1 \psi_1]^*= \lambda_1^* \lambda_1^*$$

$$\therefore\; [\hat A \psi_1]^* = \lambda_1 \psi_1^*\dotsm(3)$$

$$\therefore\; [ \hat A\psi_2]^*= \lambda_2 \psi_2^*\dotsm(4)$$

Multiplying equation (4) by \(\psi_1|0 and integrating, we get,

$$\int[\hat A\psi_2]^*\psi_1 dx= \lambda_2\int \psi^* \psi_1 dx\dotsm(5)$$

Since, \(\hat A\) is Hermitian operator,

$$\int [ \hat A\psi_2]^*\psi_1 dx= \int [ \hat A \psi_1] \psi_2^* dx\dotsm(6)$$

Using equation (6) in (5), we get,

$$\int \psi_2^*(\hat A \psi_1)dx = \lambda_2\int \psi_2^*\psi_1 dx$$

From (1)

$$\int \psi_2^* ( \lambda_1\psi_1) dx= \lambda_2 \int \psi_2^* \psi_1 dx$$

$$or,\; \lambda_1- \lambda_2\int_{-\infty}^\infty \psi_2^*\psi_1dx=0$$

Since \(\lambda_1- \lambda_2\ne 0\)

$$\therefore\; \int \psi_2^* \psi_1 dx=0\; OR\; \int \psi_2 \psi_1^* dx=0$$

Thefore \(\psi_1\) and \(\psi_2\) are orthogonal.

Similarly, we can show that \(\int\psi_1^* \psi_2 dx=0\) Proved.

Parity Operator \([\hat \pi] \):

Parity is defined as the behaviour of system upon space reflection or space inversion about origin. The parity of a system is represented by parity operator.

Parity operator \((\hat \pi\)) changes the sign of space co-ordinate of system what comes after it.

Mathematically,

$$\hat\pi \psi(\vec r) = \psi(-\vec r) \dotsm(1)$$

$$\\hat \pi \psi(x,y,z)= \psi(-x,-y,-z)\dotsm(2)$$

where,

\(\vec r\)= Position vector of a point = \(\hat i x + \hat j y + \hat k z\)

(x,y,z)= Co-ordinate of a point.

Properties of parity operator \((\hat\pi\)):

1. The eigen values of parity operator is always real. [ i.e. either +1 or -1]

\(\Rightarrow\) Proof:

Let \(\lambda\) be the eigen value of \(\hat \pi\) over wave function \(\psi(r)\)

From eigen value equation \([\hat \pi \psi(r)= \lambda \psi(r)]\dotsm(1)\)

Again,

Operating equation (1) by \(\hat \pi\) and using value of equation (1)

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

$$\therefore \hat \pi^2\psi(r)=\lambda^2\psi(r)\dotsm(2)$$

From the definition of parity operator.

$$\hat \pi \psi(\vec r) = \psi(-\vec r)\dotsm(3)$$

Again, operating above equation by \(\hat \pi\) on both sides and using definition of parity operator.

$$\hat \pi^2 \psi(r)= \hat \pi \psi(-r)$$

$$\hat \pi^2 \psi(r)= 1\psi(r)\dotsm(4)$$

From equation (2) and (4)

$$\lambda^2\psi(r)= 1\psi(r)$$

$$\Rightarrow \lambda^2=1$$

$$\Rightarrow \lambda=\pm 1$$

$$\Rightarrow \lambda= \;either \;+1\\ or\; -1$$

therefore, the eigen values of parity operator is always real.

When \(\lambda= +1\),

$$\hat \pi\psi(r)=\lambda \psi(r)\;\;\;[From\;1}

$$[\psi(-r)= \psi(r)]\dotsm(6)$$

Where \(\psi(r)\) is even function.

If eigen values of parity operator is +1 then the function is said to be even function of position.

When \(\lambda=-1\)

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

$$[\psi(-r)= -\psi(r)]\dotsm(7)$$

\(\psi(r)\) having eigen value -1 of \(\hat \pi\) is said to have odd parity or odd function of position.

(2) Parity operator is linear operator:

\(\Rightarrow\) Proof:

Let \(\psi(r)\) be the linear combination of two wave function \(\psi_1(r)\) and \(\psi_2(r)\).

$$i.e \psi(r)= a\psi_2(r) + b\psi_2(r)\dotsm(1)$$

Where, a and b are complex coefficients.

Now,

$$\hat \pi[ a\psi_1(r)+ b\psi_2(r)]= a\psi_2(-r)+ b\psi_2(-r)\; [ From \; Definition ]$$

$$\therefore\; \hat \pi[ a \psi_1(r)+ b\psi_2(r)] = a\hat \pi\psi_1(r)+ b\hat \pi \psi_2(r)\dotsm(2)$$

Here, \(\hat \pi\) is linear operator.

(3) Parity operator is Hermitian operator:

\(\Rightarrow\) Proof: Let \(\psi_1(x)\) and \(\psi_2(x)\) be any two well defined wave function of parity operator \(\pi\).

$$\int_{-\infty}^\infty [ \hat \pi \psi_1(x)]^*\psi_2(x) dx= \int_{-\infty}^\infty [ \psi_1(-x)]^* \psi_2(x)dx\dotsm(2)$$

Put\( y=-x\Rightarrow x=-y\rightarrow dx= -dy\)

When, \(x=-\infty \Rightarrow y= + \infty\)

\(x= +\infty\rightarrow y=-\infty\)

From (1)

$$\int_{-\infty}^\infty [\hat \pi \psi_1(x)]^*\psi_2(x)dx= \int_{+\infty}^{-\infty}[\psi_1 (y)]^*\psi_2(-y)(-dy)$$

$$=\int_{-\infty}^\infty [\psi_1(y)]^* \psi_2(-y)dy$$

$$= \int_{-\infty}^\infty[\psi_1(y)]^*(\hat \pi \psi_2(y))dy\dotsm(2)$$

$$\int_{-\infty}^\infty [ \hat \pi \psi_1(x)]^* \psi_2(x)dx= \int_{-\infty}^\infty [ \psi_1(x)]^* [ \hat \pi \psi_2(x)]dx\dotsm(3)$$

Hence, parity operator \((\hat \pi\)) is Hermitian operator.

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.