Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974) Lecture 3

Topics Today Infinite Square Well Harmonic Oscillator

Infinite Square Well L V(x) (Probability of finding a particle at x>L and x<0 is zero) 3

Particle in a Box                                                   which requires 4

Normalization, Infinite Square Well Wave Function For the infinite square well, the probability must be equal to one for finding it within the well. The condition for normalization is then                                                                                                                             5

Solution of Infinite Square Well Stationary States of Infinite Square Well: Most general solution: By Dirichlet Theorem: 6

Solution of Infinite Square Well By Dirichlet Theorem: Please refer to Example 2.2 in Griffith. Sum of probabilities = 1 Probability of getting a particular energy is independent of time: 7

Example 2.2 (Griffith) A particle in an infinite square well has the initial wave function For some constant A. Outside the well, of course, Ψ = 0, Find Ψ(x,t).

Example 2.2 (Griffith): Solution Normalize Ψ(x,0) to determine A:

Example 2.2 (Griffith): Solution

The Harmonic Oscillator Using Hooke’s Law: Potential Energy: Schrodinger Equation:

Harmonic Oscillator: Analytical Method Approximate solution: Is chosen. It satisfies the requirement of going to zero at infinity. Therefore, it is possible to normalize the wavefunction.

The wavefunctions for the quantum harmonic oscillator contain the Gaussian form which allows them to satisfy the necessary boundary conditions at infinity. In the wavefunction associated with a given value of the quantum number n, the Gaussian is multiplied by a polynomial of order n called a Hermite polynomial. The expressions are simplified by making the substitution The general formula for the normalized wavefunctions is = Hermite Polynomials

The general formula for the normalized wavefunctions is Gaussian Function = Hermite Polynomials

Quantum Harmonic Oscillator Wave Functions

The general solution to the Schrodinger equation leads to a sequence of evenly spaced energy levels characterized by a quantum number n.

Quantum Harmonic Oscillator: Wavefunctions The Schrodinger equation for a harmonic oscillator may be solved to give the wavefunctions illustrated below.

Problem 1 A particle in the infinite square well has the initial wave function (a) Sketch Ψ(x,0) and determine constant A. (b) Find Ψ(x,t). (c) What is the probability that a measurement of the energy would yield the value E1? (d) Find the expectation value of energy.

PROBLEM 2 A particle of mass m moves in a harmonic oscillator potential. The particle is in the first excited state Calculate <x> for this particle. Calculate <p> for this particle. (iii) Calculate <p2> for this particle. (iv) At what positions are you most likely to find the particle? At what position are you least likely to find the particle?

Problem 3 A particle in a harmonic oscillator potential starts out in the state Y(x,0) = A[3y0(x) +4y1(x)] Find A. Construct Y(x,t) and IY(x,t)I2. Find <x> and <p>. Check that Ehrenfest’s theorem holds for this wavefunction. If you measured the energy of this particle, what values might you get?