Modern Physics Unit QM3 The Schrödinger Equation

Overview: This unit introduces the Schrödinger wave equation used to describe quantum mechanical systems such as atoms. This is a differential equation for the wave function, subject to certain boundary conditions. The square of this wave function gives the relative probability for finding a particle. This unit also introduces quantum mechanical operators for physical quantities such a position and momentum and shows how these operators can be used to calculate average values for the quantities from the wave functions.

Prerequisite: QM2

R. Eisberg and R. Resnick Quantum Mechanics of Atoms, Solids, Nuclei and Particles (2nd Ed.) , Chapt. 5 - Schrödinger's Theory of Quantum Mechanics. Sec. 1-8; Chapt. 6 - Solutions of Time-Independent Schrödinger Equation, Sec. 2-4,7,8.

Videotape: There is a videtape of a lecture by Prof. Mohan Kalelkar providing an explanation of the key concepts and problem-solving techniques for this unit. If you wish to view the tape during class ask your instructor to set you up in the nearby video room. This tape can also be viewed in the Math and Science Learning Center (MSLC) by asking at the reception desk for Physics 323 Tape QM3 on the Schrodinger Equation.

The video may also be viewed online here

COMMENT:

You have learned that particles have wave-like properties, with the wavelength related inversely to the momentum via the de Broglie relation. For a particle in a non-uniform potential, however, the classical momentum changes with position and the question arises how to interpret the wave nature in this case. Schrödinger suggested the differential equation which now bears his name for the "wave function".

The wavefunction has wave-like properties and its absolute magnitude square (the wave function is in general a complex number) determines the probability per volume for finding the particle assocaiated with the wavefuction.

For determining quantities like the average position or average momentum of the particle one has only to multiply the product of the corresponding "operator" with the wave function by the complex conjugate of the wavefunction and then integrate over all space.

After completing this unit you should understand::

1. The heuristic derivation of Schrödinger's equation based upon the conservation of energy.
2. How the time-independent Schrödinger equation is obtained from the time-dependent equation by the separation of variables.
3. How energy quantization arises.
4. How acceptable wave functions and the corresponding discrete (quantized) values for the energy are obtained for the simple case of a one dimensional infinite square well potential.
5. How the probability distribution P(x,t) is related to the wave function psi(x,t).
6. Average or expectation values of a measurable quantity (observable).
7. What operators are and how they are used to find expectation values.
8. How to differentiate between the average of a set of measurements ("expecation values") and the individual measured values themselves ("eigenvalues").
9. How to sketch matter waves in different potential regions for both bound and free states.

Problems:

Chapt. 5: Questions 3,4,5,8,12,16,17,21 and Problems 1,23.

Chapt. 6: Questions 15,19,23 .

Also do the following problem: Let an atom exist in 2 states of different energyes E_1 and E_2 . Let its wave function be Psi(x,t) = [Psi_1(x,t) + Psi_2(x,t)]/sqrt(2), where Psi_1(x,t)=exp(-i omega_1 t) psi_1(x,t) and Psi_2(x,t)=exp(-i omega_2 t) psi_2(x,t) .

1. If the energy operator is i hbar partial time derivative, what is , the average value of E?
2. Does change with time? Does , the expectation value of x, depend upon time? Explain any differences between these two cases.

Sketch a wave function (real part or magnitude) of indicated energy E in the potentials shown.