2023-09-29 Quantum Physics Lecture 9

Last time: Expectation values and Gaussian distributions, as well as Heisenberg Uncertainty Principle #qp1physlecture

Superposition of waves and time evolution

Recall superposition of waves.

$${\psi }_1+{\psi }_2=?$$

In previous lectures we have defined

$$\psi (x)=\int A(k)\sin (kx)dk$$

So, $A(k)=?$

In general,

$$\psi (x)=\sum c_n\sin \left(\frac{n\pi x}{L}\right)$$

The previous portion($\sin \left(\frac{n\pi x}{L}\right)$) of the above formula is also the basis for our Functional Vector Space? Wave superposition gives rise through time dependence. Example:

$$\Psi (x,t)=\frac{1}{\sqrt{2}}{\Psi }_1(x,t)+\frac{1}{\sqrt{2}}{\Psi }_2(x,t)$$

Solving this by including the appropriate factor of $e^{-i{E}_{n}t/\hslash }$ we get.

$$e^{-i{E}_{n}t/\hslash }\left[\frac{1}{\sqrt{2}}{\psi }_1+\frac{e^{-i{E}_{n}t/\hslash }}{\sqrt{2}}\right]$$

And the probability density is therefore given by

$$|\Psi (x,t){|}^2={\Psi }^{\ast }\Psi =\frac{1}{2}{\psi }_1^2\frac{1}{2}{\psi }_2^2+{\psi }_2{\psi }_1\cos (E_2-E_1)t/\hslash$$

Recall:

\Psi_{n}(x,t)=\sum c_{n}\varphi_{n}(x)e^{-iE_{n}t/\hbar} $$$ @ $t=0$ This is also the Fourier series expansion? Need to talk to him in office hours about the rest of this section. I don't understand wtf is a Fourier coefficient.