PH 1130 Term C, 2001 STUDY GUIDE #6

QUANTUM MECHANICS

I. The Wavefunction Revisited:

The goal of physics is to understand the universe around us. This means we want to describe and predict phenomena in a consistent manner. For example, in Classical Mechanics (CM), its position and velocity specify a particle’s motion or the evolution of the particles state. So, it’s future is predicted by Newton’s Law, F = ma. This is called an Equation of Motion. Now, in Quantum Mechanics (QM) we need an Equation of Motion to describe the future of the wavefunction Φ(r,t) in the same way.

Review Sections 8.2 and 8.3 carefully.

Much of the math for this material can be intense, so try and follow it carefully. However, keep in mind that our goal is to use the math to understand the concepts so you won’t be asked to reproduce any of these derivations.

Remember, for the classical wave picture, what is being described is the position of each part of the string at any time. For the quantum wave picture, what is being described is the wavefunction (which describes the state the particle is in) which has a position part and a time part:

Φ(r,t) = φ(r)exp(-iωt) , with the energy given by E = hω.

Recall that the probability density is |Φ(r,t)|2 = |φ(r)|2|exp(-iωt)|2 = |φ(r)|2

NOTE: the sum of all probabilities must add up to one (the normalization condition).

So, quantum standing waves = probability density that is independent of time! (also called a stationary state).

Review and Study Section 8.4, a particle in a box.

II. Schrödinger’s Equation:

Differential equations underlay much of the math used in physics (recall the definitions of acceleration, velocity, and position and how they show up in F = ma). Schrödinger argued (but could not prove!) that an appropriate differential equation for QM should look like:

∂ 2ψ 2m = []U (x) − E ψ (x) , in one-dimension (only x-component) ∂x 2 h 2 ∂ 2ψ ∂ 2ψ ∂ 2ψ 2m + + = []U − E ψ , in three-dimensions ∂x 2 ∂y 2 ∂z 2 h 2

where U is the potential energy and E is the total energy (thus U – E = - KE, the kinetic energy).

Note: there is no way to prove this equation. It is a hypothesis that has passed every experimental test in the 60 years since it was introduced. One of its greatest achievement is its use in Atomic Physics in explaining energy levels of atoms.

Review Examples 8.2 and 8.3 and Study carefully Section 8.9, the Simple Harmonic Oscillator (SHO) to understand where the derivation of the energy levels comes from.

III Radioactive decay. Tunneling:

Study Sec. 13.1 - 13.4. Try to follow Sec. 13.9. Your goal here is to understand the concept of tunneling. Refer to the lecture material.

A. In radioactivity or in nuclear reactions, conservation laws must be obeyed: energy, angular momentum, and electric charge are the well-known ones. In addition, the number of electrons plus the number of neutrinos minus the number of antineutrinos is a constant; that is called "lepton conservation". Finally the number of protons plus the number of neutrons is constant; that is called "baryon conservation".

B. There are three principal types of naturally occurring radioactivity, governed by the strong, weak, and electromagnetic interactions respectively. In alpha-decay an alpha particle (4He nucleus) is emitted from a heavy nucleus, leaving a nucleus with both Z and N reduced by two; e.g.,

238 234 92U146 - 90 Th 144 + a

In beta-decay an electron or positron (beta-particle) and an antineutrino or a neutrino are created, and Z of the nucleus changes by one. The mass number remains unchanged. For example,

31H2 - 32He1 + e + v

40 40 + 19K21 - 18A22 + e + v

In gamma-decay, a nucleus in an excited state emits a photon, and goes to a lower energy state:

7Li* → 7Li + γ

The process is the same as the de-excitation of an atom, except that the energies are normally much larger.

IV. Pauli Exclusion Principle:

This skips around a little but refer to Section 11.4 for details. This Section uses Periodic Table to illustrate the Exclusion principle. I will give a simpler discussion of the Exclusion Principle in the lecture.

The Pauli exclusion principle describes a fundamental property of all particles whose spin quantum numbers are odd half-integers. That includes electrons, protons, and neutrons. We first specify all quantum numbers of a quantum-mechanical state; in the case of an atom, this would include n, l, m, s, and ms. The exclusion principle then guarantees that no two electrons have identical sets of quantum numbers. This principle guarantees that atoms behave as the periodic chart describes, and it also accounts for the chemical properties of all the elements. If electrons had integral spin, h or zero, for example, there would be no atomic structure, and no chemistry; because particles with integral spin don't obey the exclusion principle, all electrons of an atom would huddle together in the same state, right on top of the nucleus!

The Pauli Exclusion Principle is an example of Fermi-Dirac Statistics while particles which don’t have to obey this principle follow Bose-Einstein Statistics. These are the two fundamental statistical descriptions in physics!

An interesting consequence of Bose-Einstein Statistics is that since all integer spin particles can collapse into the ground-state (lowest energy state), the entire collection can be described by a single wavefunction. Which means that this collection can act like one, huge, particle! (called Bose-Einstein Condensation).