Difference between revisions of "Introduction to Quantum Mechanics/Chapter 1"

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== Notes ==
 
== Notes ==
 +
 +
=== The Wave Equation ===
  
 
The Schrodinger Wave Equation:
 
The Schrodinger Wave Equation:
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* <math>V(x,t)</math>, the potential.
 
* <math>V(x,t)</math>, the potential.
  
−
This is probably the first time you're seeing something like this, so don't let it overhwelm you.
+
=== 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.
 +
 
 +
:<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>\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>
  
−
The wave function itself is somewhat meaningless. We'll talk about how we can get physical things we are familiar with, like position and momentum, out of it.
+
<math>\sigma</math> is the standard deviation, <math>\sigma^2</math> is the variance.
  
−
This is a second-order differential equation. They are not trivial to solve, particular for non-trivial V.
+
:<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>

Revision as of 16:00, 3 May 2012

Introduction to Quantum Mechanics

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.

<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>\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>

<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>

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