Lecture 7- Eigenvalues and Eigenfunctions

Review

De Broglie wavelength

$$\lambda =\frac{h}{p}=\frac{2\pi }{k}$$ $$\omega =\frac{\hslash }{E}$$

Light radiates at the frequency of the slosh

New notation!

$$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+V(\psi )\approx i\hslash \frac{d\psi }{dt}$$

Wave function must be continuous and normalizable Continuous is insisted upon, should be normalizable (unless its $A{e}^{i(kx-\omega t)}$ )

Smoothity of functions is kind of required. Unless we have infinite potential energy Infinite square well

Special case

$$V=V(x)$$ $$\frac{d^2\psi }{d{x}^2}+(E-V)\psi =0$$

Observables $⟨q⟩$

$$=\int {\psi }^{\ast }{q}_{op}\psi dx$$

Time independent Schrodinger equation

$$f(t)=e^{i\omega t}$$

Operators

$$\mathcal{H}=-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V$$ $$E_{op}=i\hslash \frac{d}{dt}$$ $$P_{op}=\frac{\hslash }{i}\frac{d}{dx}$$ $$x_{op}=x$$ $$\psi (x)\approx \sqrt{\frac{1}{2\pi }}\int A(k)e^{ikx}dk$$ $$A(k)\approx \sqrt{\frac{1}{2\pi }}\int \psi (x)e^{-ikx}dx$$

probability current

$$j\approx \frac{i\hslash }{m}\left({\psi }^{\ast }\frac{d\psi }{dx}-\psi \frac{d\psi \ast }{dx}\right)$$ $${\sin }^{2}x\approx \frac{1}{2}-\frac{1}{2}\cos 2x$$