Properties of Hermitian operator

Properties of Hermitian operators:

(1) Sum of any two like Hermitian is also Hermitian.

Let $\hat A$ and $\hat B$ be any two like Hermitian operators.

Sum of two operators, $\hat C= \hat A+ \hat B$

We have to show that $\hat C$ is also Hermitian

$\Rightarrow$ Let $\psi(x)$ be a valid wave function.

Now,

$$\int_{-\infty}^\infty \psi^*(\hat A\psi)dx= \int_{-\infty}^\infty (\hat A \psi)^* \psi dx \dotsm(1)$$

$[ \hat A$] is Hermition.

and

$$\int_{-\infty}^\infty \psi^*(\hat B\psi) dx= \int_{-\infty}^\infty ( \hat B \psi)^*\psi dx\dotsm(2)$$

$[\hat B$] is Hermitian

Adding (1) and (2), we get,

$$\int_{-\infty}^\infty \psi^*(\hat A\psi)dx+ \int_{-\infty}^\infty \psi^*(\hat B\psi )dx= \int_{-\infty}^\infty (\hat A\psi)^*\psi dx+ \int_{-\infty}^\infty (\hat B \psi)^*\psi dx$$

$$or,\;\; \int_{-\infty}^\infty \psi^*[\hat A\psi+ \hat B\psi]dx= \int_{-\infty}^\infty [ (\hat A\psi)^* + (\hat B\psi)^*]\psi dx$$

$$or,\;\; \int_{-\infty}^\infty [ (\hat A+\hat B) \psi]dx= \int_{-\infty}^\infty [(\hat A\psi + \hat B\psi)^*]\psi dx$$

$$or,\;\; \int_{-\infty}^\infty \psi^*(\hat c\psi)dx= \int_{-\infty}^\infty [ (\hat c \psi)]^*\psi dx\dotsm(3)$$

Where, $\hat c= \hat A+ \hat B$ is Hermitian.

(2) Eigen value of a Hermitian operator is always real:

$\Rightarrow$

Let $\lambda$ be the eigen value of a Hermitian operator $\hat A$ ove wave function $\psi(x)$.

The eigen value equation of \(\hat A\0 is

$$\hat A \psi(x)= \lambda \psi(x)\dotsm(1)$$

We have to show that $\lambda$ is real i.e. $\lambda^*= \lambda$.

Taking complex conjugate of equation (1)

$$[ \hat A \psi(x)]^*= [ \lambda \psi(x)]^*$$

$$or,\;\; [ \hat A\psi(x)]^*= \lambda^* \psi^*(x)\dotsm(2)$$

Multiplying equation (2) by $\psi(x)$ from right and integrating

$$\int_{-\infty}^\infty [ \hat A\psi(x)]^*\psi(x) dx= \lambda^*\int_{-\infty}^\infty \psi^*(x) \psi(x)dx\dotsm(3)$$

Since, $\hat A$ is Hermitian.

$$\int_{-\infty}^\infty \psi^*(x)[\hat A\psi(x)]dx= \int_{-\infty}^\infty[\hat A\psi(x)]^* \psi(x)dx\dotsm(4)$$

Using (4) in (3), we get,

$$or,\;\; \int_{-\infty}^\infty \psi^*(x)[\hat A\psi(x)]dx= \lambda^*\int_{-\infty}^\infty \psi^*(x) \psi(x)dx$$

Using equation (1)

$$or,\;\; \int_{-\infty}^\infty \psi^*(x) [\lambda \psi(x)]dx= \lambda^*\int_{-\infty}^\infty \psi^*(x)\psi(x)dx$$

$$or,\;\; \lambda\int_{-\infty}^\infty \psi^*(x)\psi(x)dx= \lambda^*\int_{-\infty}^\infty \psi^*(x)\psi(x)dx$$

$$or,\;\; (\lambda-\lambda^*) \int_{-\infty}^\infty \psi^*(x)\psi(x)dx=0$$

Since, $$\int_{-\infty}^\infty \psi^*(x)\psi(x)dx\ne 0$$

$$\lambda- \lambda^*=0$$

$$\therefore\; \lambda= \lambda^*$$

Hence, eigen value of a Hermitian operator is always real.

3. the product of two Hermitian operator is Hermitian only if they commute.

i.e. If $\hat A$ and $\hat B$ are any two Hermitian operators then their product $\hat A\hat B$ is Hermitian only if $[\hat A, \hat B$] =0.

$\Rightarrow$ Proof:

As $\hat A$ is Hermitian:

$$\int_{-\infty}^\infty (\hat A\psi)^*\psi dx= \int_{-\infty}^\infty \psi^*(\hat A\psi)dx\dotsm(1)$$

As $\hat B$ is Hermitian:

$$\int_{-\infty}^\infty (\hat B\psi)^* \psi dx= \int_{-\infty}^\infty \psi^*(\hat B \psi)dx\dotsm(2)$$

The product of $\hat A$ and $\hat B$ is $\hat A\hat B$.

Now, $$\int_{-\infty}^\infty [ \hat A \hat B\psi]^*\psi dx= \int_{-\infty}^\infty[\hat A(\hat B\psi)^*\psi dx$$

$$=\int_{-\infty}^\infty [ \hat B \psi]^* [ \hat A \psi] dx$$

$\therefore \hat A$ is Hermitian.

$$=\int_{-\infty}^\infty \psi^*[ \hat B ( \hat A \psi) ] dx$$

$\therefore \hat B$ is alos Hermitian.

$$=\int _{-\infty}^\infty \psi^*[ \hat B\hat A \psi] dx$$

$$or,\;\; \int_{-\infty}^\infty [ \hat A\hat B\psi]^* \psi dx = \int_{-\infty}^\infty \psi^* [ \hat A\hat B\psi]dx\; only\; if \; \hat A \hat B= \hat B \hat A$$

$$\rightarrow [ \hat A, \hat B] =0$$

Hence product of the two Hermitian operator is Hermitian only if they commute.

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.