Chapter (6) Introduction to Quantum Mechanics

 is a single valued function, continuous, and finite every where

Example(6.1) Normalizing the Wavefunction The initial wavefunction of a particle is given as  (x,o)= C exp(  x  /x o ), where C and x o are constants. 1- Sketch  function. 2-Find C in terms of x o such that (x, 0) is normalized. Solution The given wavefunction is symmetric, decaying exponentially from the origin in either direction, as shown in Figure,The decay length x o represents the distance over which the wave amplitude is diminished by the factor 1/e from its maximum value  (0, 0) = C. The normalization requirement is

Because the integrand is unchanged when x changes sign we may evaluate the integral over the whole axis as twice that over the half-axis x > 0, where  x  = x. Then

EXAMPLE 6.2 Calculate the probability that the particle in the preceding example will be found in the interval solution or about 86.5%, independent of x o.

Wavefunction For A Free Particle A free particle is one subject to no force The wave number k and frequency  of free particle matter waves are given by the de Broglie relations For non relativistic particle  is related to k as The wave function for a free particle can be represented by a plane wave This is an oscillation with wavenumber k, frequency , and amplitude A (travelling wave). If each plane wave constituting the packet is assumed to propagate independently of the others according to equation (2), the packet at any time is (2) From (1)

Example Find the wavefunction  (x, 0) that results from taking the function Solution Write, where C and  are constants

or For a free particle U(x)=0,, Schrödinger equation : Which is the total energy

The Particle In A Box ( Infinite Square Well ) The particle can never be found outside, i.e.  (x)=0 outside. Inside the box U(x)=0 Schrödinger equation is 0 L  U  (1) k is the wavenumber of oscillation, the solution to this equation is. Boundary condition: from the boundary conditions one obtains A and B n=1,2,….

Using Relation (1) becomes Which shows that the particle energies are quantized and gives the energy levels Ground state or zero point energy Exited state Notice that E=0 is not allowed, that is the particle can never be at rest. E 1 is zero-point energy > 0

Wave function:- Return to Probability:-

Normalization:-

The Quantum Oscillator Consider the problem of a particle subject to a linear restoring force F = - kx. Here x is the displacement of the particle from equilibrium (x =0) and k is the spring constant. *The potential energy is The angular frequency, The total energy, The quantum oscillator is described by the potential energy In the Schrödinger equation

The solution may be in the form Where C 0 and  are constants, using this solution, we get

Comparing with (1), we get This is the ground state energy With Eigen function:

Normalization of the ground state wavefunction

Exited state Comparing with (1) This is the first excited state Eigen energy So the energy levels for the harmonic oscillator is. The separation between any two levels is n=0,1,2,… E 0 is the ground state energy level

Expectation values A particle described by the wavefunction  may occupy various places x with probability given by  2 We have several values of x, we need the average value of x The average value of x, written  x , is called the expectation value and defined as Which gives the average position of a particle. The average value of any function f(x) is In quantum mechanics the standered derivation,  x, is called the uncertainty in position, and given by The degree to which particle position is fuzzy is given by the magnitude of  x. the position is sharp only if  x=0

EXAMPLE: Location of a Particle in a Box Compute the average position  x  and the quantum uncertainty in this value,  x, for the particle in a box, assuming it is in the ground state. Solution The ground state wavefunction is with n =1 for the ground state. The average position is calculated as