CHAPTER 2

Schrodinger Theory of Quantum Mechanics

a)For Free particle A particle which is not acted upon by any external force is called a free particle. Potential energy of a free particle is zero(V=0).The total energy is only given by kinetic energy of the particle.

Time-Dependent Schrodinger equation for a free particle in one dimension is

And in three dimension

Where is the Laplacian operator.

TIME DEPENDENT SCHRODINGER EQUATION

b) For Forced particle:Time-dependent Schrodinger equation for a particle subjected to some force is given as:In one dimensionIn three dimension

CONTINUED….

Time-independent Schrodinger equation for a one dimensional system isIn three dimensions

Time-independent Schrodinger equation is also known as Steady State equation.

TIME-INDEPENDENT SCHRODINGER EQUATION

Wave function (x) and d(x)/dx must satisfy the following conditions:1. (x) and d(x)/dx must be single-valued.2. (x) and d(x)/dx must be finite everywhere.3. (x) and d(x)/dx must be continuous everywhere.4. (x) must vanish at infinity i.e, 0 as x.

ACCEPTABLE OR WELL-BEHAVED WAVE FUNCTION

ORTHONORMAL WAVE FUNCTIONS Normalized wave function- A wave function is said to be normalized if it satisfy following condition

Orthogonal wave function- Two wave functions are said to be orthogonal if it satisfy following condition

Orthonormal Functions- which satisfy the following condition

where mn =1 for m=n and mn =0 for m n.

An operator is a mathematical instruction which acts on a given function changes it into another function or the same wave function multiplied by some constant real or complex. In quantum mechanics with every dynamical variable like position, momentum , energy etc. there exists an operator.Linear momentum operator- Energy Operator –Hamiltonian Operator-

OPERATOR

An operator A is said to be Hermitian, if it satisfies following condition

Properties of Hermitian Operator: The Hermitian operator have real eigenvalues. Two Eigen functions of Hermitian Operator, belonging to different eigen values are orthogonal.

HERMITIAN OPERATOR

It states that: The average motion of a wave packet agrees completely with the motion of the classical particle i.e. wave packet moves like a classical particles. Using the equations of motion from classical mechanics for expectation value of position and momentum for a wave packet we can writea)The time rate change of average distance is velocityb)The time rate change of average velocity is negative of potential gradient or force.

EHRENFEST’S THEOREM

If an operator acts on a wave function and we get the same wave function multiplied by some constant, then the wave function is known as eigen function and constant is known as eigen value.where may be real or complex. In this case

(x, t) is eigen function and is eigen value of operator.

EIGEN FUNCTIONS AND EIGENVALUES

The expectation value of any observable is defined as

If the wave function is normalized, the denominator in the above definition reduces to unity.Commutator of two operators-Two operators are said to commute with each other if their commutator is zero.The operators which do not commute with each other are known as “canonically conjugate ” variables.Position-Momentum Commutator-Time-Energy Commutator-

EXPECTATION VALUES OR AVERAGE VALUES OF DYNAMIC VARIABLES