Responsible party: Schrödinger's Dog

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Q_A

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…)
• Born's postulate and Yuichi's discussion on how Born'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, and look at <math> |E|^2 \prop</math> Intensity <math>\prop</math> Energy density <math>\prop</math> number of particle <math>\prop</math> probability. From this, we can see that the square of a classical wave can be interpreted as a probability, which Yuichi speculates Born used to interpret <math> |\Psi|^2 </math> to be the probability density.

Using the fact, which is proved in the next section by Yuichi, we find that the normalization factor holds for all time t. Griffths shows this on page 13 and 14 of the text. I will reiterate the proof here. To show the normalization factor A is time independent, we need to show that <math> \int_{-\infty}^{\infty}\Psi^*\Psi dx </math> is independent of t. If so, then <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx =0 </math>. Bringing in the total derivative with respect t, we get a partial derivative in the integral expression before to get: <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t}(\Psi^*\Psi)dx</math>. Differentiating and using the Schrodinger equation we get, <math> \frac{\partial}{\partial t}|\Psi|^2=\Psi*\frac{\partial \Psi}{\partial t}+\frac{\partial \Psi*}{\partial t}\Psi= \frac{i\hbar}{2m}(\Psi*\frac{\partial^2 \Psi}{\partial x^2}+\frac{\partial^2 \Psi*}{\partial x^2}\Psi)= \frac{\partial}{\partial x}[\frac{i\hbar}{2m}(\Psi*\frac{\partial \Psi}{\partial x}+\frac{\partial \Psi*}{\partial x}\Psi)]. </math> Putting this into <math>\frac{d}{dt} \int_{-\infty}^{\infty}|\Psi^2| dx </math>, using the Fundamental Theorem of Calculus, and the condition that <math>\Psi(+\infty)=\Psi(-\infty)=0 </math>, we get that <math>\frac{d}{dt} \int_{-\infty}^{\infty}|\Psi^2| dx = \frac{i\hbar}{2m}(\Psi*\frac{\partial \Psi}{\partial x}+\frac{\partial \Psi*}{\partial x}\Psi)</math> going from <math> -\infty </math> to <math>+\infty </math>. Using the condition <math>\Psi(+\infty)=\Psi(-\infty)=0 </math>, we find that <math> \frac{d}{dt} \int_{-\infty}^{\infty}\Psi^*\Psi dx =0 </math>. This means our integral is constant(independent of time), and hence if <math> \Psi </math> is normalized at t=0 and stays normalized for all future time. QED.

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. Yuichi used the definition of a derivative and contour plots to show that this statements was true. This is easy to do when you let <math> g(t)= \int|\Psi|^2 </math>, we can show equivalence of the two integrals.

This concluded the lecture.

Yuichi

Clarification on this point. What we want to show is that <math>\frac{d}{dt} \int_{-\infty}^{\infty} f(x,t) dx=\int_{-\infty}^{\infty}\frac{\partial}{\partial t} f(x,t) dx</math>.
By the definition of differentiation, <math>\frac{d}{dt} \int_{-\infty}^{\infty} f(x,t) dx=\lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} f(x,t+\Delta t) dx - \int_{-\infty}^{\infty} f(x,t) dx}{\Delta t} </math>.

In order to get the difference between the two terms in the numerator of the RHS, it may be convenient to express <math>f(x,t+\Delta t)</math> in terms of <math>f(x,t)</math>. If we CHOOSE to compare these two terms for the same value of x* (note that this is up to me), we find that <math>f(x,t+\Delta t) = f(x,t)+\frac{\partial f(x,t)}{\partial t}\Delta t</math>. By subbing this into the above expression with a limit, we find that <math>\lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} f(x,t+\Delta t) dx - \int_{-\infty}^{\infty} f(x,t) dx}{\Delta t} = \lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t}\Delta t dx}{\Delta t} = \int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t} dx</math>.

*Even though it's silly to choose different value of x by <math>\Delta x</math>, then <math>f(x+\Delta x,t+\Delta t) = f(x,t)+\frac{\partial f(x,t)}{\partial t}\Delta t+ \frac{\partial f(x,t)}{\partial x}\Delta x</math>.

Then, <math>\lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} f(x+\Delta x,t+\Delta t) dx - \int_{-\infty}^{\infty} f(x,t) dx}{\Delta t} = \lim_{\Delta t\rightarrow 0} \frac{\int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t}\Delta t dx + \frac{\partial f(x,t)}{\partial x}\Delta x dx}{\Delta t} = \int_{-\infty}^{\infty} \frac{\partial f(x,t)}{\partial t} dx + \Delta x[f(x,t)]_{-\infty}^{\infty}</math>. Provided that <math>f(x,t)</math> is a normalizable function, the last term should be zero.

Thanks Yuichi, this makes a lot more sense!

S.D.: I wasn't really clear on the last proof, but feel that it isn't a important detail. Besides that, the rest of the lecture was great!

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