Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

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HERMITIAN OPERATOR

According to second postulate to every observable in classical mechanics, there corresponds an Hermitian operator in quantum mechanics.

A more general definition of Hermitian operator is:

If wave function are same,Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

If any operator A satisfy above condition is a Hermitian operator.

Eigen values of Hermitian operator are real and eigen function of Hermitian operator are orthogonal.

 

Hermitian Operators:

Quantum mechanical operators are Hermitian operators. AS Hermitian operator  obeys the relation

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev        Hermitian Operator - Atomic Structure Chemistry Notes | EduRevProperty of Hermitian operator

Where dq is volume element. For example ‘τ’ represents the Cartesian coordinates of a particle that can move in three dimensions, the integral is a three-fold integral and dq stands for (dxdydz).

The symbol Hermitian Operator - Atomic Structure Chemistry Notes | EduRev f* denotes the complex conjugate of the function f, and Hermitian Operator - Atomic Structure Chemistry Notes | EduRev A* denotes the complex conjugate of the operator A.  Complex quantities are surveyed briefly in Appendix B. If z is a complex quantity it can be written .

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevz = x + iy

Where the real quantity ‘x’ is called the real part of ‘z’ and the real quantity of ‘y’ is called the imaginary part of ‘z’ . The complex conjugate of any complex number. Function, or operator is obtained by changing the sign of its imaginary part:

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

A real quantity or a real operator is equal to its complex conjugate, and imaginary quantity or an imaginary operator is equal to the negative of its complex conjugate.

Thus. We can write definition of Hermitian operator as an operator that satisfies the relation

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev ...........(1)

Where Hermitian Operator - Atomic Structure Chemistry Notes | EduRev is any well-behaved function.

Problem: Show that momentum operator is Hermitian

Sol. Momentum operator p = –ihd/dx

Substitution into equation (1)

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

And          Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Thus, we see that Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev  does, indeed, satisfy the equation (1). Therefore, the momentum operator is a Hermitian operator.

If follows from the definition of Hermitian operators that an operator  is Hermitian. When it is its own Hermitian adjoint – that is Hermitian Operator - Atomic Structure Chemistry Notes | EduRev . On the other hand an operator  is its complex conjugate – that is Hermitian Operator - Atomic Structure Chemistry Notes | EduRev at = at

Here are some relationships concerning Hermitian adjoints:

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

  

Problem: Show that Px is Hermitian by using concept of Hermitian adjoint.

Soln. AS we know any operator is hermitian only if    

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev      Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

AVERAGE VALUE OR EXPECTATIONN VALUE

If function Hermitian Operator - Atomic Structure Chemistry Notes | EduRev ψ is normalized than Hermitian Operator - Atomic Structure Chemistry Notes | EduRev Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

SCHRODINGER EQUATION

The operator corresponding to the energy E is called the Hamiltonian operator and is represented by the symbol HOP. Thus we have.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev......(5)
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev      ...................(6)

Now according to equation (2) we have.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev       .................(7)

Where E is the eigenvalue of the Hamiltonian operator HOP a quantity which has a precise value for a given state. Thus, Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev    ....................(8)

It is called schrodinger wave equation.


APPLICATION OF SCHRODINGER EQUATION:

Particle in a box → Translational energy → Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Simple Harmonic Oscillation → Vibrational energy → Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Rigid rotor → Rotational energy → Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Hydrogen atom → Electronic energy → Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev Potential Energy

  

PARTICLE IN A ONE DIMENSIONAL BOX

In this problem a particle of mass is placed in a one dimensional box of length/particle is free to move. Box have infinite high walls.

It is assumed that the potential energy of particle is zero every where inside the box.

That is potential          V(x)=0

Thus the one dimensional schrodinger equation, we have.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Neglecting E is comparison to ∞, we have.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

That is Hermitian Operator - Atomic Structure Chemistry Notes | EduRev outside the box. This means that the particle cannot exist outside the region 0 < x < 1. Within the box, the schrodinger equation for the motion of the particle takes the form.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev                   .................(2)
Where, Hermitian Operator - Atomic Structure Chemistry Notes | EduRev  .................(3)

A general solution of equation (2) is given by.

 Hermitian Operator - Atomic Structure Chemistry Notes | EduRev.................(2)

Where A and B are constant.

Only those function which satisfy the boundary conditions if Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev will be acceptable wave functions for ψ = 0 at x = 0 Hermitian Operator - Atomic Structure Chemistry Notes | EduRev equation (4) becomes.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

The above expression will be true only where αl is an integral multiple of π that is,

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev  ...........(6)

When n can have only integral values of 1,2,3,……….A value of n=0 is eliminated since it leads to α=0 or Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRevevery where within the box. Substituting α from equation (6) in equation (5) we get.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev..................(7)
Now from equation (3) we have.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev......................(8)

Substituting the value of α from equation (6) we have, Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev.........(9)

In this case, n represents the quantum number.

So energy difference between the two successive energy levels.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

As we knowHermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev
so Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

 

ZERO POINT ENERGY (Ground state energy):

For ground state energy we will put n = 1 in the equation.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev      n = 1Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

 

PARTICLE IN TWO DIMENSIONAL BOX

For 2-D box wave function Hermitian Operator - Atomic Structure Chemistry Notes | EduRevψ  will dependent upon two independent variable x and y and Hermitian Operator - Atomic Structure Chemistry Notes | EduRev is the multiplication of both function. Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Schrodinger wave equation for free particle for 2-D box:

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Dividing both side by X(x), Y(y).

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
 

The term Hermitian Operator - Atomic Structure Chemistry Notes | EduRev and Hermitian Operator - Atomic Structure Chemistry Notes | EduRev in the above equation is a constant quality. L.H.S must also be a constant quantity and x, y both are independent on the other term and each is equal to a constant quantity.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev......................(1)
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev......................(2)
 

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevSolution of equation (1), Hermitian Operator - Atomic Structure Chemistry Notes | EduRev  Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Solution of equation (2), Hermitian Operator - Atomic Structure Chemistry Notes | EduRev Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

DEGENERACY

More than one state with equal energy
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev   Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

Zero Point Energy:
nx = 1 n= 1           Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
nx = 1 n= 2        Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

PARTICLE IN THREE DIMENSIONAL BOX

Now we will discuss the motion of particle of mass m in three dimensional box. As in one dimensional box in three dimensional box also, the potential energy is zero with in the box and infinite outside the box.

So three dimensional schrodinger equation.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Where the function Hermitian Operator - Atomic Structure Chemistry Notes | EduRevψ  will depend upon three independent variable x, y, z to solve the above equation we write the function Hermitian Operator - Atomic Structure Chemistry Notes | EduRev ψ  as the product of three wave function.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev

Where, Hermitian Operator - Atomic Structure Chemistry Notes | EduRevX(x), Y(y), Z(z) are three function.

Put this value in equation (1).

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev.......(2)

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev .......(3)

Dividing by XYZ, we have.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev.......(4)

The term Hermitian Operator - Atomic Structure Chemistry Notes | EduRevα2 in the above equation is a constant quantity. Hence the sum of the three terms on the left hand side of equation (4) must also be a constant quantity. If we change the value of x (or y or z) keeping the other two variables constants even then the above constancy has to be satisfied. This is possible only when each term is independent of the other term and each is equal to a constant quantity so that the sum of three constant is equal to Hermitian Operator - Atomic Structure Chemistry Notes | EduRev .α2

So we write.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev.......(5)
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev.......(6)
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev .......(7)

Where,         Hermitian Operator - Atomic Structure Chemistry Notes | EduRev     Hermitian Operator - Atomic Structure Chemistry Notes | EduRev    .......(8)
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev         .......(9)
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev        .......(10)                  Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

With,               Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev .......(11) 

And               E = Ex + Ey + Ez  Hermitian Operator - Atomic Structure Chemistry Notes | EduRev .......(12) 

Now we have three separate equations to be solved each of them has a form of one-dimensional box. Thus the normalized wave function of a three-dimensional box is

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev
Hermitian Operator - Atomic Structure Chemistry Notes | EduRev..........(13)

The constant Hermitian Operator - Atomic Structure Chemistry Notes | EduRev will be given by.

Hermitian Operator - Atomic Structure Chemistry Notes | EduRev..........(14)

And total energy s Hermitian Operator - Atomic Structure Chemistry Notes | EduRev       E = Ex + Ey + Ez  Hermitian Operator - Atomic Structure Chemistry Notes | EduRevHermitian Operator - Atomic Structure Chemistry Notes | EduRev        ..........(15)

There are three quantum numbers one each for energy every degree of freedom.

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