angular momentum operator in matrix form

Laplacian in spherical coordinate

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NMR

NMR is a technique to detect the state of nuclear spin. a similar technique on electron spin is call ESR ( electron spin resonance)

The principle of NMR is simple.

1. apply a B-field, and the spin will align with it due to interaction with surrounding and precessing along the B-field with Larmor frequency. the time for the spin align with the field is call T1, longitudinal relaxation time. 
2. Then, we send a pule perpendicular to the B-field, it usually a radio frequency pulse. the frequency is determined by the resonance frequency, which is same as the Larmor frequency. the function of this pulse is from the B-field of it and this perpendicular B-field with perturb the spin and flip it 90 degrees. 
3. when the spin are rotate at 90 degrees with the static B-field, it will generate a strong enough signal around the coil. ( which is the same coil to generate the pule ) and this signal is called NMR signal. 
4. since the spins are not isolate, when it interact with environment, they will go back and align with the static B-field. the time for this is called T2, transverse relaxation time. 

by analyzing the T1 and T2 and also Larmor frequency, we can known the spin, the magnetization, the structure of the sample, the chemical element, the chemical formula, and alot many others thing by different kinds of techniques.

For nuclear physics, the use of NMR is for understand the nuclear spin. for example, the polarization of the spin.

Changing of frame

sometimes, i will confuse on changing frame, especially between rotating frame and lab frame.

WKB approximation

I was scared by this term once before. in fact, don't panic, it is easy. Let me explain.

i just copy what written in Introduction to Quantum Mechanics by David Griffiths (1995) Chapter 8.

The approx. can be applied when the potential is varies slowly compare the wavelength of the wave function. when it expressed in Exp[ i k x], wavelength = 2 pi / k, when it expressed in Exp[ - kapper x ], wavelength = 1/kapper.

in general, the wavefunction can be expressed as amplitude and phase:

A[x] Exp[ i phase[x] ], A[x] and phase[x] are real function.

sub this into the Schrödinger equation ( Psi ''[x] = - { 2 m / hb (E - V[x])} Psi[x] ). and separate the imaginary part and real part.

The imaginary part is can be simplified as:

D[ A^2 phase' ] = 0 => A = Const. /Sqrt[ phase'[x] ]

The real part is

A''[x] = ( phase'[x]^2 -  2 m / hb (E - V[x]) ) A[x]

we use the approx. that A''[x] = 0 since it varies slowly.

Thus,

phase'[x] =  Sqrt[2 m / hb (E - V[x])]
=>
phase[x] = Integrate[ Sqrt[2 m / hb (E - V[x])], x]

and the solution is, ( if we use p[x] = Sqrt[2 m / hb (E - V[x])] )

Psi [x] = Const. /Sqrt[ p[x] ] Exp[ i Integrate[ p[x] ] ]

Simple! but one thing should keep in mind that, the WKB approx is not OK when Energy = potential.

This tell you, the phase part of the wave function is equal the square of the area of the different of Energy and the Potential.

when the energy is smaller then the potential, than, the wavefunction is under decay.

one direct application of WKB approxi is on the Tunneling effect.
if the potential is large enough, so, the transmittance is dominated by the decay, Thus, the probability of the tunneling is equal to Exp[- 2 Sqrt[ 2m / hb] Sqrt[area between V[x] and Energy] ]. Therefore, when we have an ugly potential, we can approx it by a rectangular potential with same area to give the similar estimation.