Introduction to Quantum Mechanics/Chapter 1

Introduction to Quantum Mechanics

• Part I: Theory
  • 1. The Wave Function
  • 2. The Time-Indepedent Schrodinger Function
  • 3. Formalism
  • 4. Quantum Mechanics in Three Dimensions
  • 5. Identical Particles
• Part II: Applications
  • 6.
  • 7.
  • 8.
  • 9.
  • 10.
  • 11.

Notes

The Wave Equation

The Schrodinger Wave Equation:

<math>i\hbar {\partial \Psi \over \partial t} = - {\hbar^2 \over 2m} {\partial^2 \Psi \over \partial x^2} + V \Psi</math>

• <math>\Psi(x,t)</math>: The wave function, varies over position and time.
• <math>i = \sqrt{-1}</math>
• <math>\hbar = {h \over 2\pi} = 1.054573 \times 10^{-34}\ J\ s</math>, Planck's Constant.
• <math>m</math>, mass of the particle.
• <math>V(x,t)</math>, the potential.

Statistical Interpretation

Square the wave function to get its position:

<math>| \Psi(x,t) |^2\ dx = \overline{\Psi}\Psi\ dx</math> is the probability of finding the particle between x and dx at time t.

Three interpretations of the wave function:

• realist: The particle was really at a particular position, the wave function simply represents the fact that we didn't know. There are hidden variables as well that are not truly random.
• orthodox: (This is correct due to Bell's Theorem) The particle wasn't anywhere. The act of measuring it causes it to choose a position from a number of possible positions. Measurements produce the particles. AKA "Copenhagen Interpretation". (Note: We still don't really know what "measurement" means.)
• agnostic: Why are we talking about the position of something we won't be able to measure? No one can really know what is happening, because if you measure it, you know what the answer is, and if you don't, you don't.

There are two types of processes:

• 'ordinary': The wave function evolves according to the wave equation.
• 'measurements': The wave function collapses to a definite point.

On measurements (borrowing from the afterward): You can think of the macroscopic world as a continuously evolving set of quantum wave functions, but since there are so many, it really doesn't make sense to think of separate realities at all. The act of measurement is really the interaction of these two systems. Heisenberg's interpretation is probably best, meaning we should call these things "events" that are recorded in an indelible record, not "measurement". No human consciousness is needed.

Probability

We take a tour of probability.

For discrete values:

<math>j</math> is a unique event.

<math>N(j)</math> is the number of unique events j.

<math>N = \sum_j N(j)</math> is the total number of events.

<math>P(j) = {N(j) \over N}</math> is the probability of even j.

<math>\sum_j P(j) = 1</math>

<math>\langle j \rangle = \sum_j j\ P(j)</math> is the average value of j.

Note that <math>\langle j^2 \rangle \ge \langle j \rangle^2</math>. If they are equal, there is no spread at all.

<math>\sigma</math> is the standard deviation, <math>\sigma^2</math> is the variance.

<math>

\begin{align} \sigma^2 &= \langle (\Delta j)^2 \rangle \\

&= \sum (j - \langle j \rangle)^2 P(j) \\
&= \langle j^2 \rangle - \langle j \rangle ^2 \\

\end{align} </math>

For continuous values:

• <math>x</math> is a possible value.
• <math>\rho(x)\ dx</math> is the probability density.
• <math>P_{ab} = \int_a^b \rho(x)\ dx</math> is the probability that x lies betwen a and b.
• <math>\int_{-\inf}^{+\inf} \rho(x)\ dx = 1</math> (something will happen.)
• <math>\langle x \rangle = \int_{-\inf}^{+\inf} x\rho(x)\ dx</math>
• <math>\langle f(x) \rangle = \int_{-\inf}^{+\inf} f(x)\rho(x)\ dx</math>
• <math>\sigma^2 = \langle x^2 \rangle - \langle x \rangle ^2</math>

Normalization

In order to make the statistical interpretation have any sense at all:

<math>\int_{-\inf}^{+\inf} | \Psi(x,t) |^2\ dx = 1</math>

non-normalizable solutions (integral is 0 or infinity) cannot represent particles and must be thrown out.

The wave function remains normalized throughout time.

Momentum

The expectation value of the position:

<math>\langle x \rangle = \int_{-\inf}^{+\inf} x|\Psi(x,t)|^2\ dx</math>

The velocity of the expectation value of the position is not the velocity!

<math>{d\ \langle x \rangle \over dt} = -{i \hbar \over m} \int_{-\inf}^{+\inf} \Psi* {\partial \Psi \over \partial x}\ dx</math>

But we can pretend it is:

<math>\langle v \rangle = {d \langle x \rangle \over dt}</math>

But we'd rather talk about momentum:

<math>\langle p \rangle = m {d \langle x \rangle \over dt}</math>

This is a convenient shorthand:

<math>\langle x \rangle = \int_{-\inf}^{+\inf} \Psi* x \Psi\ dx</math>

<math>\langle p \rangle = \int_{-\inf}^{+\inf} \Psi* {\hbar \over i}{\partial \over \partial t} \Psi\ dx</math>

Thus, the operator:

• <math>x</math> represents position
• <math>{\hbar \over i}{\partial \over \partial t}</math> represents momentum
• <math>Q(x, {\hbar \over i}{\partial \over \partial t})</math> represents any quantity <math>Q(x,p)</math>.