HW 5: Schrodinger Equation and Operators - Quantum Mechanics for Chemistry

"Where did we get that (equation) from? Nowhere! It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.” — Richard Feynman

Q1¶

A. Arrive at Time independent Schrodinger equation. Start first plugging this standing wave $\psi(x,t) = \psi(x) cos(\omega t)$ into classical wave equation. Next express wavelength in terms of momentum via De Broglie relation $\lambda=\frac{h}{p}$ . Finally express momenta in terms of the total energy of the system $E=\frac{p^2}{2m}+{V}$
- Hint: This is done in the book. But try to practice your hand at doing the steps.
B. Arive at Time dependent Schrodinger equation. Start by taking derivatives of a simple traveling periodic wave $\psi(x,t)=e^{i(kx-\omega t)}$ with respect to time and space. Hint :
- Note that by taking one time derivative you get $i\omega$ in front which can be related to Energy E. How? by recalling planck’s formula $E=h\nu = \hbar \omega$ This is why we see $\hbar=h/2\pi$ frequently.
- When taking two derivatives with respect to x you get $-k^2$ . This can also be related to energy $E$ by recalling definition of a wave-number $k=\frac{2\pi}{\lambda}$ and planck’s formula.
- This procedure shows you the reason why in Schordinger equation you have second derivative with respect to space and single derivative with respect to time.

Q2¶

Evaluate $g=\hat{A}f$ given the following pairs of operator $\hat{A}$ and function $f$

$\hat{A}=\sqrt{....}$	$f=x^4$
$\hat{A} =\frac{d^3}{dx^3}+x^3$	$f=e^{-ax}$
$\hat{A} = \int^{1}_0 dx$	$f=x^3-2x+3$
$\hat{A}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$	$x^3y^2z^4$

Q3¶

Determine which of the opeartors are linear. You can do this by tesing to see weather the linearity condition is satisdied: $\hat{L}(c_1 f_1+c_2 f_2)=c_1\hat{L}f_1+c_2\hat{L}f_2$

$\hat{L}f=\sqrt{f}$
$\hat{L}f=f^{*}$
$\hat{L}f=0$
$\hat{L}f=\frac{1}{f}$
$\hat{L}f=f(0)$
$\hat{L}f=ln f$

Q4¶

Show that $\psi(x,t)=e^{i(kx-\omega t)}$ is an eigenfunction for operators $\hat{A_1}=\frac{d^2}{dt^2}$ and $\hat{A_2}=\frac{d^2}{dx^2}$
Write down eigenvalues of the two operators.
Are the two operators linear?
Is the sum of the two opearators linear $\hat{A_1}+\hat{A_2}$
Would linear combination of linear operators be a linear operator itself?

Q5¶

Write out the operator $\hat{A}^2f$ acting on some arbitrary function $f$ for the following cases:

$\hat{A}=\frac{d^2}{dx^2}$
$\hat{A}=\frac{d}{dx}+x$