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# Hermitian Operator - Atomic Structure Chemistry Notes | EduRev

## Physical Chemistry

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## Chemistry : 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,

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 Â obeys the relation

Property 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  f* denotes the complex conjugate of the function f, and  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 .

z = 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:

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

...........(1)

Where  is any well-behaved function.

Problem: Show that momentum operator is Hermitian

Sol. Momentum operator p = –ihd/dx

Substitution into equation (1)

And

Thus, we see that   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 Â is Hermitian. When it is its own Hermitian adjoint – that is  . On the other hand an operator Â is its complex conjugate – that is  at = at

Here are some relationships concerning Hermitian adjoints:

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

Soln. AS we know any operator is hermitian only if

AVERAGE VALUE OR EXPECTATIONN VALUE

If function  ψ is normalized than

SCHRODINGER EQUATION

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

......(5)
...................(6)

Now according to equation (2) we have.

.................(7)

Where E is the eigenvalue of the Hamiltonian operator HOP a quantity which has a precise value for a given state. Thus,

....................(8)

It is called schrodinger wave equation.

APPLICATION OF SCHRODINGER EQUATION:

Particle in a box → Translational energy →

Simple Harmonic Oscillation → Vibrational energy →

Rigid rotor → Rotational energy →

Hydrogen atom → Electronic energy →  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.

Neglecting E is comparison to ∞, we have.

That is  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.

.................(2)
Where,   .................(3)

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

.................(2)

Where A and B are constant.

Only those function which satisfy the boundary conditions if  will be acceptable wave functions for ψ = 0 at x = 0  equation (4) becomes.

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

...........(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 every where within the box. Substituting α from equation (6) in equation (5) we get.

..................(7)
Now from equation (3) we have.

......................(8)

Substituting the value of α from equation (6) we have,

.........(9)

In this case, n represents the quantum number.

So energy difference between the two successive energy levels.

As we know
so

ZERO POINT ENERGY (Ground state energy):

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

n = 1

PARTICLE IN TWO DIMENSIONAL BOX

For 2-D box wave function ψ  will dependent upon two independent variable x and y and  is the multiplication of both function.

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

The term  and  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.

......................(1)
......................(2)

Solution of equation (1),

Solution of equation (2),

DEGENERACY

More than one state with equal energy

Zero Point Energy:
nx = 1 n= 1
nx = 1 n= 2

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.

Where the function ψ  will depend upon three independent variable x, y, z to solve the above equation we write the function  ψ  as the product of three wave function.

Where, X(x), Y(y), Z(z) are three function.

Put this value in equation (1).

.......(2)

.......(3)

Dividing by XYZ, we have.

.......(4)

The term α2 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  .α2

So we write.

.......(5)
.......(6)
.......(7)

Where,                  .......(8)
.......(9)
.......(10)

With,                .......(11)

And               E = Ex + Ey + Ez   .......(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

..........(13)

The constant  will be given by.

..........(14)

And total energy s        E = Ex + Ey + Ez          ..........(15)

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

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## Physical Chemistry

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