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. T IME D EPENDENT S CHRODINGER E QUATION

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

Time-independent Schrodinger equation for a one dimensional system is In three dimensions Time-independent Schrodinger equation is also known as Steady State equation. T IME -I NDEPENDENT S CHRODINGER E QUATION

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 . A CCEPTABLE OR W ELL -B EHAVED WAVE FUNCTION

O RTHONORMAL 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- O PERATOR

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. H ERMITIAN O PERATOR

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 write a)The time rate change of average distance is velocity b)The time rate change of average velocity is negative of potential gradient or force. E HRENFEST ’ S T HEOREM

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. E IGEN FUNCTIONS AND E IGENVALUES

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- E XPECTATION V ALUES OR A VERAGE V ALUES OF D YNAMIC V ARIABLES