Responsible party: Schrödinger's Dog, Devlin

The main points of today's lecture:

• discussion and interpretation in quantum mechanics (i.e. <p>, <math>\Psi^*\Psi dx=P(x)dx</math>, etc…)
• Bohm's postulate and Yuichi's discussion on how Bohm's may have used classical ideas to come up with <math>\Psi^*\Psi dx=P(x)dx</math>
• Why we use <math>|\Psi|^2</math>, and not just <math>\Psi</math>, or some linear combination of <math>\Psi</math> for the probability

*normalization for t=0 and how the normalization works for time t * How partial derivatives and ordinary derivatives are used and the justification of the equation <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t}(\Psi^*\Psi)dx=0</math>

In the beginning of lecture, Yuichi discussed how <math> m\frac{d<x>}{dt}</math> is interpreted to be <p> and how we get this interpretation from the classical idea that p=mv. Yuichi then went on to discuss how <K> is found from the interpretation that <math> <K>=\frac{<p>^2}{2m}=-\frac{\hbar}{2m}(\frac{\partial}{\partial x})^2</math> and how this can be generalized to an Q(x,p) (i.e. L, a, etc…), where Q expectation value would be <math> <Q(x,p)> = \int \Psi* Q(x,-i\hbar \frac{\partial}{\partial x}) \Psi dx </math>. Later in chapter 3, we will see that the last result is valid.

Yuichi then moves on to talk about why <math>|\Psi|^2</math> and showed that working with a <math>\Psi<0</math> or even worse <math>\Psi</math> is complex, then we aren't able to have physical meaning of <math>\Psi</math>. What gets rid of this problem is <math>|\Psi|^2</math>, which gets rid of a negative and/or complex wave function.

We then discussed how Born may have come up with the interpretation that Psi squared is the Probability Density. Yuichi told us that Born probably looked to classical physics, where he came up with this idea that <math>\Psi^*\Psi dx=P(x)dx</math>. Yuichi explained that when we looked at electromagnetic wave E,

Using the fact, which is proved in the next section by Yuichi,

At the end of lecture, Yuichi goes on to prove equation 1.21 or <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t}(\Psi^*\Psi)dx=0</math>, which Griffths fail to show rigorously. He first starts to talk about the 2nd inequality in the last equation and what it means to take a partial derivative and why we are allowed to hold x fix in <math> (\frac{\partial}{\partial t} f(x,t))_x </math> when we do put in the integrand.

(will finish 9/12)