Hermitian Operators

We discussed about Hermitian operator. In physics, an operator is a function over a space of physical states to another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are a very useful tool in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Summary

We discussed about Hermitian operator. In physics, an operator is a function over a space of physical states to another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are a very useful tool in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Things to Remember

  1. All the obserable quantities in Quantum Mechanics are associated with corresponding operators The expectation values of the quantity is always real and the corresponding operator associated with real sign values is called Hermitian Operator.

  2. $$or, \int_\tau (\hat A\psi)^* \psi d\tau= \int_\tau \psi^*(\hat A\psi) d\tau$$\dotsm(3)$$

    Any operator \(\hat A\) which satisfy equation (3) is known as Hermitian operator. In general for any two normalized wavefuntion, we can write

    $$\int_\tau(\hat A\psi_1)^* \psi_2 d\tau= \int_\tau \psi_1^*(\hat A\psi_2)d\tau\dotsm(4)$$

    Then \(\hat A\) is Hermitian.

  3. $$\therefore\; [\hat L_z, \hat L_+]= \hbar \hat L_+$$
  4. $$[\hat L_z, \hat L_{\pm}]= \pm\hbar L_{\pm}$$

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Hermitian Operators

Hermitian Operators

Hermitian operators:

All the obserable quantities in Quantum Mechanics are associated with corresponding operators The expectation values of the quantity is always real and the corresponding operator associated with real sign values is called Hermitian Operator.

Let \I\hat A\) be an operator associated with quantity A. Over normalized state \(\psi(r)\), the expectation value of A is written as,

$$<A>= \int_{\tau} \psi^* (\hat A\psi)d\tau\dotsm(1)$$

Since <A> is real.

We can write,

$$<A>^*= <A>\dotsm(2)$$

Using equation (1) in (2)

$$or,\;\; \biggl[ \int_{\tau} \psi^*(\hat A\psi)d\tau\biggr]^*= \int_\tau \psi^*(\hat A\psi)d\tau$$

$$or,\;\int_\tau [\psi^*(\hat A\psi)]^*d\tau= \int_\tau \psi^* (\hat A\psi)d\tau$$

$$or, \int_\tau (\hat A\psi)^* \psi d\tau= \int_\tau \psi^*(\hat A\psi) d\tau$$\dotsm(3)$$

Any operator \(\hat A\) which satisfy equation (3) is known as Hermitian operator. In general for any two normalized wavefuntion, we can write

$$\int_\tau(\hat A\psi_1)^* \psi_2 d\tau= \int_\tau \psi_1^*(\hat A\psi_2)d\tau\dotsm(4)$$

Then \(\hat A\) is Hermitian.

Proof:

$$[\hat L_+, \hat L_-]$$

$$=[\hat L_x+ i\hat L_y, \hat L_x- i\hat L_y]$$

$$=[ \hat L_x, \hat L_x]- i[\hat L_x, \hat L_y+ \hat i [ \hat L_y, \hat L_x]- i^2[ \hat L_y, \hat L_y]$$

$$=-i(i\hbar)\hat L_z+i(-i\hbar)\hat L_z$$

$$=-i^2\hbar\hat L_z- i^2\hbar \hat L_z$$

$$=2\hbar\hat L_z$$

$$\therefore\; [\hat L_-, \hat L_+]= -2\hbar \hat L_z$$

Show that \([\hat L_z, \hat L_+]= \hbar \hat L_+\)

$$[\hat L_z, \hat L_-]= -\hbar \hat L_-$$

$$OR$$

$$[\hat L_z, \hat L_{\pm}]= \pm\hbar \hat L_{\pm}$$

Proof: (a)

$$[\hat L_z, \hat L_+]$$

$$= [\hat L_z, \hat L_z+ i \hat L_y]$$

$$=[\hat L_z, \hat L_x]+i[\hat L_z, \hat L_y]$$

$$=i\hbar\hat L_y+ i(-i\hbar)\hat L_x$$

$$= i\hbar \hat L_y+ \hbar \hat L_x$$

$$= \hat (\hat L_x+ i\hat L_y)$$

$$\therefore\; [\hat L_z, \hat L_+]= \hbar \hat L_+\dotsm(1)$$

$$Proved$$

Proof: (b)

$$[\hat L_z, \hat L_-]$$

$$=[\hat L_z, \hat L_x- i\hat L_y]$$

$$=[\hat L_z, \hat L_x]- i[\hat L_z, \hat L_y]$$

$$=i\hbar \hat L_y-i(-i\hbar)\hat L_x$$

$$=i\hbar \hat L_y- \hbar \hat L_x$$

$$=-\hbar(\hat L_x- i\hat L_y)$$

$$=-\hbar \hat L_-\dotsm(2)$$

$$Proved$$

Combining (1) and (2) we get,

$$[\hat L_z, \hat L_{\pm}]= \pm\hbar L_{\pm}$$

$$Proved$$

1. \(\psi(x), \psi(r,t)\)

Usual Notation Dirac Notation

1. \(\psi(x),\psi(r,t)\)

 \(|\psi>\)= ket-state or, ket vector.
2. \(\psi^*(x),\psi^*(r,t) \(<\psi|\)= Bra-state, or Bra-vector
3. Condition of normalization\(\int_{-\infty}^\infty \psi^*(x)\psi(x)dx=1\) \(<\psi|\psi>\)=1 Condition of Normalization

4. Hermitian (\( |\hat A\))

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

 \(<\psi|\hat A\psi>= < \hat A\psi|\psi>\)
5. \(\int_\tau (\hat a\psi_1)^* \psi_2 d\tau= \int_\tau \psi_1^* (\hat A \psi_2 d\tau\) \(< \hat A\psi_1| \psi_2>= < \psi_1|\hat A\psi_2>\)

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.

Lesson

Operator formalism in Quantum mechanics

Subject

Physics

Grade

Bachelor of Science

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