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How to incorporate spin into the wavefunction?

by HomogenousCow
Tags: incorporate, spin, wavefunction
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HomogenousCow
#1
Nov18-12, 06:13 AM
P: 366
I'm currently reading the Griffith book and he dosen't really explain this, do i just miltiply the spin kets with the wave function??
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dextercioby
#2
Nov18-12, 07:54 AM
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I'd go for less mathematics this time:

Yes, the spin dof do not interfere with the coordinate dof which are used in the normal wavefunction,(= a spin 0 wavefunction), that's why there's a multiplication of the spin ket (a 1-column matrix with 2, 3, 4, etc. components) with a wavefunction depending on x,y,z (or px,py,pz if working in the momentum representation).
andrien
#3
Nov18-12, 10:02 AM
P: 1,020
In non relativistic quantum mechanics spin is introduced in a rather ad hoc manner and by solving the schrodinger eqn for say hydrogen atom one can introduce all quantum numbers but spin.Spin was introduced as another degree of freedom because it can not be obtained from schrodinger eqn.However pauli introduced it for spin 1/2 by noting that p2 can be written as (σ.p)(σ.p),but it is still artificial.The logical way of introducing it was done by dirac for spin 1/2.

dextercioby
#4
Nov18-12, 10:17 AM
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How to incorporate spin into the wavefunction?

Quote Quote by andrien View Post
In non relativistic quantum mechanics spin is introduced in a rather ad hoc manner and by solving the schrodinger eqn for say hydrogen atom one can introduce all quantum numbers but spin.Spin was introduced as another degree of freedom because it can not be obtained from schrodinger eqn.However pauli introduced it for spin 1/2 by noting that p2 can be written as (σ.p)(σ.p),but it is still artificial.The logical way of introducing it was done by dirac for spin 1/2.
This is wrong. Spin is not introduced ad-hoc, it's a logical consequence of the Galilean theory, as is in specially relativistic quantum theory. The work of LÚvy-Leblond in the 1960's should not be discarded.

Nonrelativistic particles and wave equations

Jean-Marc LÚvy-Leblond; 286-311
Commun. math. Phys. 6, 286—311 (1967)

Page 289, to be precise.

http://projecteuclid.org/DPubS?servi...cmp/1103840276
andrien
#5
Nov18-12, 01:17 PM
P: 1,020
I was saying it in historical way,nevertheless the reference you have given accepts the dirac original view to make a non-relativistic version of dirac eqn and it is not a logical consequence of Galilean theory because it nevertheless assumes linearity in both time and space derivative which was the original point of dirac in his paper.So i will still call it some way of fixing it,rather than some original motivation on Galilean relativity.
Bill_K
#6
Nov18-12, 05:17 PM
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Thanks
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The Pauli Equation, 1927, is quadratic, not linear, consequently it bears a strong resemblance to the Schrodinger Equation but not Dirac. Furthermore it was not advertised as a "nonrelativistic version" of the Dirac Equation, which was introduced in 1928, a year later.
dextercioby
#7
Nov19-12, 11:49 AM
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Quote Quote by andrien View Post
I was saying it in historical way,nevertheless the reference you have given accepts the dirac original view to make a non-relativistic version of dirac eqn and it is not a logical consequence of Galilean theory because it nevertheless assumes linearity in both time and space derivative which was the original point of dirac in his paper.So i will still call it some way of fixing it,rather than some original motivation on Galilean relativity.
We're speaking of different things. The reference I've given was meant for the idea that
spin < ----- Galilean relativity, not only that spin < ------ Lorentz/PoincarÚ/Einstein/Minkowski relativity (thing which has been known since 1928). The issue with Galilei relativistic equations vs Lorentz relativistic equations is totally different and I was not addressing it (it's actually a very broad subject).


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