1. 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

2. Topics Today
Infinite Square Well
Harmonic Oscillator

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

4. Particle in a Box
                                                 
which requires 4

5. 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

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

7. 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

8. 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).

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

10. Example 2.2 (Griffith): Solution

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

12. 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.

13. 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

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

15. Quantum Harmonic Oscillator Wave Functions

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

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

18. 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.

19. 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?

20. 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?