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.
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
Where is any well-behaved function.
Problem: Show that momentum operator is Hermitian
Sol. Momentum operator p = –ihd/dx
Substitution into equation (1)
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
The operator corresponding to the energy E is called the Hamiltonian operator and is represented by the symbol HOP. Thus we have.
Now according to equation (2) we have.
Where E is the eigenvalue of the Hamiltonian operator HOP a quantity which has a precise value for a given state. Thus,
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.
A general solution of equation (2) is given by.
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,
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.
Now from equation (3) we have.
Substituting the value of α from equation (6) we have,
In this case, n represents the quantum number.
So energy difference between the two successive energy levels.
As we know
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.
Schrodinger wave equation for free particle for 2-D box:
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.
Solution of equation (1),
Solution of equation (2),
More than one state with equal energy
Zero Point Energy:
nx = 1 ny = 1
nx = 1 ny = 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).
Dividing by XYZ, we have.
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.
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
The constant will be given by.
And total energy s E = Ex + Ey + Ez = ..........(15)
There are three quantum numbers one each for energy every degree of freedom.