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Question		Answer
De broglie wavelength equation start learning		p=h/lambda; p = mv
v_ph phase velocity start learning		v_ph = w/k
v_g group velocity start learning		d/dk (w)
angular momentum start learning		L=rp
Gaussian integral inf{x^2n * e^-ax^2 dx start learning		(2n-1)!!!/(2a)^n * sqrt(pi/a)
0toinf{e^-ax dx start learning		1/a
0toinf{x^n * e^-ax dx start learning		n!/ a^(n+1)
Probability current density start learning		j = ħ/m * Im(phi* * dphi/dx) = ħ / 2mi (phi* * dphi/dx - phi * dphi*/dx)
PHI(x,t) for stationary state start learning		=phi(x)*e^(-iEt/ħ)
probability density of PHI start learning		p(x,t) = \|PHI(x,t)\|^2
continuity equation start learning		dp/dt = dj/dx
normalization wave function phi(x) start learning		inf{\|phi(x)\|^2 dx = 1
Momentum operator p start learning		= -iħ d/dx
operator expectation value start learning		goes between phi* and phi
variance delta operator A start learning		(delta A)^2 = <A^2> - <A>^2
heisenberg uncertainty principle start learning		delta x * delta p >= ħ/2
ħ start learning		=h/2pi
operator hermitian when start learning		A=Across
inflection (turning) point start learning		when E - V(x_t) =0
For stationary wave functions start learning		phi’’ = 2’/ħ^2 * (V-E) * phi
<f\|g> start learning		inf{f*(x) g(x) dx
\|phi1> and \|phi2> ortho when start learning		<phi1\|phi2> = 0
Cauchy schwarz start learning		\|<phi1\|phi2>\|^2 <= <phi1\|phi1><phi2\|phi2>
[x, p] start learning		=iħ
[p, x] start learning		-iħ
stationary schrodinger equTion start learning		-ħ^2/2m phi’’ + Vphi = E phi
for phi to be an energy eigenstate start learning		E must be constant independent of x
energy level of oscillator start learning		E_n = ħOMEGA(n+1/2) and V(x) = 1/2 m ONEGA^2 x^2
normalization of phi matrix start learning		requires phi cross * phi = 1
P_x(+) start learning		= \|<x+\|phi>\|^2
ritz principle start learning		<phi\|H\|phi> >= E_0
standard hydrogen 1s wavefunction start learning		phi(r) = 1/sqrt(pia_0^3) e^-r/a_0
<r> start learning		4pi * 0toinf{r^3 \|phi\|^2 dr
phi 2s hydrogen start learning		in sqrt add * 32 all times (2-r/a_0) * e^-r/2a_0
hydrogen atom behaves like start learning		boson since even number of fermions (2 electrons and 2 protons)
for a free particle E start learning		E = ħw = ħ^2*k^2 / 2m
For plane-wave amplitudes reflective R start learning		= \|jreflected\| / \|jincoming\| for equal k, R = \|B\|^2/\|A\|^2

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