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Larmor Procession

Magnetic moment (mu) :

this is a magnet by angular momentum of charge or spin. its value is:

mu = gamma S ( in case of spin ) = gamma L ( in case of angular momentum )

where gamma is (g-factor) x ( Bohr magneton ) / ( h_bar [ in case that the S or L is not using natural unit ] )

the g-factor is a dimensionless number, which reflect the environment of the spin, for orbital angular momentum, g = 1.

Bohr magneton = electron charge x h_bar / ( 2 mass), since different particle has different mass, their Bohr magneton value are different. electron is the lightest particle, so, it has largest value on Bohr magneton.

J-coupling ( i am not sure it is correct to say so)

since the spin and angular momentum are mathematical equal. they give the same effect when react with magnetic field, thus, we use a new term:

J = L + S

mu = gamma J

in this case, since the g-factor of L and S are different, and when we combine with different particles, the Bohr magnetons are different, in that case, special treatment is needed. but it does not change the principle.

Larmor frequency:

When applied a magnetic field on a magnetic moment, the field will cause the moment process around the axis of the field. the precession frequency is called Larmor frequency.

the procession can be understood in classical way or QM way.

Classical way:

the change of angular momentum is equal to the applied torque. and the torque is equal to the magnetic moment  cross product with the magnetic field.

d J / d t = mu x B = gamma J x B

solving gives the procession frequency is :

w = - gamma B

the minus sign is very important, it indicated that the J is precessing by left hand rule.

QM way:


The Schrödinger equation is :

i  d/dt |Psi > = H |Psi >

H is the Hamiltonian = - mu . B = - gamma J.B = - gamma B Jz = w Jz

Jz is an operator.

the solution is for |Psi> is

|Psi ( t ) > = Exp[ - i w Jz ] |Psi ( t = 0 ) >

for up state coefficient a1 and down state coefficient a2,

solving a1 and a2 gives:

a1 = Exp [ - i w t ]
a2 = Exp [ + i w  t ]

Thus, in QM point of view, the state does not change but only the phase.

For not using natural unit, we knew, w h_bar = Energy.

Spin

( this is just a draft, not organized )

Spin is a intrinsics property of elementary particle, such as electron, proton, and even photon. Intrinsics means it is a built-in property, like mass, charge. Which extrinsic properties are speed, momentum.

Spin is a vector or tensor quality while charge and mass are scaler.

Spin can react with magnetic field, like charge reacts with electric field or mass react with force produce acceleration. Thus, spin is like a bar-magnet inside particle, counter part of charge.

The magnitude of spin is half integer or integer of reduced Planck's constant h_bar or hb. Particles with half integer of spin are classified as Fermion, and those with integer spin are Boson. they follow different statistic while interact together, thus, this creates different physics for different group.

we are not going to the mathematic description this time.

the effect of spin causes the magnetic moment, that's why it react with magnetic field. the other thing that creates magnetic moment is angular momentum for charge particle, like electron orbiting around nucleus. So, both spin and angular momentum can be imagined as a little magnet, thus, they can interact, in physics, we call the interaction between spin and angular momentum is coupling. for example, spin-orbital coupling, spin-spin coupling, etc..

when the spin interact with external magnetic field, it will process around the magnetic field with Larmor frequency. and the direction of the spin while undergoes procession can only be certain angle. for spin half, like electron or proton. there are only 2 directions, and we called it up and down.

Hall effect

It is a short review.

The hall probe is perpendicular to the B field( pointing up) and have a current I passing through ( going forward ).

Due to the Lorentz force. The positron is moving to right and accumulate. The accumulating charge creates a electric force to the left to against further positron accumulate. The magnetic force will be balanced by the electric force. Due to the electric force, there is associated voltage across the hall probe. This voltage is called hall voltage.

FB= e v B = Vh d

Where e is positron charge, v is speed of positron, B is the magnetic field, Vh is the Hall voltage and d is the distance across the hall probe.

The current I is

I = A n e v

Where A is the cross section area of the hall probe, n is density of the positron carrier, v is the positron velocity.

Thus,

Vh = I B / ( V n e )

Where V is the volume of the hall probe. But the V n is equal to the total number of positron N.

Vh = I B / ( N e )

Which is to say, the hall voltage is proportional to the magnetic field.

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.

terminology

this part is for people who confess and poor in english :P

Nucleus : (noun) is the core of element, which contain protons and neutrons

Nuclei : (noun) many of nucleus

Nuclear : (adj) to relate something to nucleus. e.g. nuclear energy, nuclear plant, nuclear phsyics

Nucleon: (noun) is the thing make the nucleus, which is a single word. a nucleon can be a proton or neutron

Nuclide: ( noun) [from google translate : A distinct kind of atom or nucleus characterized by a specific number of protons and neutrons ] (what??)

Slide rule

um... i don't think it is good to post in here.

moved to : http://gucciproxy.blogspot.com/2011/01/slide-rule.html

and http://gucciproxy.blogspot.com/2011/01/slide-rule-2.html

Hydrogen Atom (Bohr Model)

OK, here is a little off track. But that is what i were learning and learned. like to share in here. and understand the concept of hydrogen is very helpful to understand the nuclear, because many ideas in nuclear physics are borrow from it, like "shell".

The interesting thing is about the energy level of Hydrogen atom. the most simple atomic system. it only contains a proton at the center, um.. almost center, and an electron moving around. well, this is the "picture". the fact is, there is no "trajectory" or locus for the electron, so technically, it is hard to say it is moving!

why i suddenly do that is because, many text books said it is easy to calculate the energy level and spectrum for it. Moreover, many famous physicists said it is easy. like Feynman, Dirac, Landau, Pauli, etc... OK, lets check how easy it is.

anyway, we follow the usual way in every text book. we put the Coulomb potential in the Schrödinger equation, change the coordinate to spherical. that is better and easy for calculation because the coulomb potential is spherical symmetric. by that mean, the momentum operator (any one don't know what is OPERATOR, the simplest explanation is : it is a function of function.) automatically separated into 2 parts : radial and angular part. The angular part can be so simple that it is the Spherical harmonic.

Thus the solution of the "wave function" of the electron, which is also the probability distribution of  the electron location, contains 2 parts as well. the radial part is not so trivial, but the angular part is so easy. and it is just Y[l,m].

if we denote the angular momentum as L, and the z component of it is Lz, thus we have,

L²Y[l,m] = l (l+1) Y[l,m]
Lz Y[l,m] = m Y[l,m]

as every quadratic operator, there are "ladder" operator for "up" and "down".

L⁺ Y[l,m] =const. Y[l,m+1]
L^- Y[l,m] = const. Y[l,m-1]

which means, the UP operator is increase the z-component by 1, the constant there does not brother us.

it is truly easy to find out the exact form of the Y[l,m] by using the ladder operator. as we know, The z component of the a VECTOR must have some maximum. so, there exist an Y[l,max] such that

L⁺ Y[l,max] = 0

since there is no more higher z-component.

by solve this equation, we can find out the exact form of Y[l,max] and sub this in to L², we can know max = l. and apply the DOWN operator, we can fins out all Y[l,m], and the normalization constant is easy to find by the normalization condition in spherical coordinate, the normalization factor is sin[theta], instead of 1 in rectangular coordinate.