Physics 6210/Spring 2007/Lecture 2
Lecture 2
Relevant sections in text: 1.2
Quantum theory of spin 1/2
We now try to give a quantum mechanical description of electron spin which matches
the experimental facts described previously.
Let us begin by stating
Physics 6210/Spring 2007/Lecture 23
Lecture 23
Relevant sections in text: 3.2, 3.5
Spin precession as a rotation
It is enlightening to return to the dynamical process of spin precession in light of our
new results on rotations. You will recall that a spin
Physics 6210/Spring 2007/Lecture 24
Lecture 24
Relevant sections in text: 3.5, 3.6
Angular momentum eigenvalues and eigenvectors (cont.)
Next we show that the eigenvalues of J 2 are non-negative and bound the magnitude
of the eigenvalues of Jz . One way t
Physics 6210/Spring 2007/Lecture 25
Lecture 25
Relevant sections in text: 3.6, 3.7
Position representation of angular momentum operators
We have seen that the position operators act on position wave functions by multiplication and the momentum operators a
Physics 6210/Spring 2007/Lecture 26
Lecture 26
Relevant sections in text: 3.6, 3.7
Two spin 1/2 systems: observables
We have constructed the 4-d Hilbert space of states for a system consisting of two
spin 1/2 particles. We built the space from the basis o
Physics 6210/Spring 2007/Lecture 27
Lecture 27
Relevant sections in text: 3.6, 3.7
Angular momentum addition in general
We can generalize our previous discussion of 2 spin 1/2 systems as follows. Suppose we
are given two angular momenta J1 and J2 (e.g., t
Physics 6210/Spring 2007/Lecture 28
Lecture 28
Relevant sections in text: 3.9
Spin correlations and quantum weirdness: The EPR argument
Recall the results of adding two spin 1/2 angular momenta. The fact that the total spin
magnitude is not compatible wit
Physics 6210/Spring 2007/Lecture 29
Lecture 29
Relevant sections in text: 3.9
Spin correlations and quantum weirdness: Spin 1/2 systems
Consider a pair of spin 1/2 particles created in a spin singlet state. (Experimentally
speaking, this can be done in a
Physics 6210/Spring 2007/Lecture 30
Lecture 30
Relevant sections in text: 3.9, 5.1
Bells theorem (cont.)
Assuming suitable hidden variables coupled with an assumption of locality to determine
the spin observables with certainty we found that correlation f
Physics 6210/Spring 2007/Lecture 31
Lecture 31
Relevant sections in text: 5.1, 5.2
Example: nite size of the atomic nucleus
One improvement on the simple particle-in-a-potential model of an atom takes account
of the fact that the atomic nucleus is not tru
Physics 6210/Spring 2007/Lecture 32
Lecture 32
Relevant sections in text: 5.2
Degenerate Perturbation Theory (cont.)
Degenerate perturbation theory leads to the following conclusions (see the text for
details of the derivation). To compute the rst-order c
Physics 6210/Spring 2007/Lecture 33
Lecture 33
Relevant sections in text: 5.2, 5.6
Example: Hyperne structure (cont.)
We are evaluating the matrix elements of the perturbation, which now takes the form
e B =
8
e p (r)
3
in the degenerate subspace spanned
Physics 6210/Spring 2007/Lecture 34
Lecture 34
Relevant sections in text: 5.6
Time-dependent perturbation theory (cont.)
We are constructing an approximation scheme for solving
i
h
i
d
cn (t) =
e h (En Em )t Vnm (t)cm (t),
dt
m
Vnm = n|V (t)|m .
For simpl
Physics 6210/Spring 2007/Lecture 35
Lecture 35
Relevant sections in text: 5.6
Fermis Golden Rule
First order perturbation theory gives the following expression for the transition probability:
(En Ei )t
4|Vni |2
P (i n, i = n) =
sin2
.
2
2
h
(En Ei )
We ha
Physics 6210/Spring 2007/Lecture 36
Lecture 36
Relevant sections in text: 5.7
What happens when you shine light on an atom?
You will have noticed that up to this point in our discussion of time-dependent perturbation theory I have assiduously avoided much
Physics 6210/Spring 2007/Lecture 37
Lecture 37
Relevant sections in text: 5.7
Electric dipole transitions
Our transition probability (to rst order in perturbation theory) is
P (i f )
1
4 2
N (f i )| nf , lf , mf |ei|f i | c nX e P |ni , li , mi |2 ,
2 2
Physics 6210/Spring 2007/Lecture 22
Lecture 22
Relevant sections in text: 3.1, 3.2
Rotations in quantum mechanics
Now we will discuss what the preceding considerations have to do with quantum mechanics. In quantum mechanics transformations in space and ti
Physics 6210/Spring 2007/Lecture 21
Lecture 21
Relevant sections in text: 3.1, 3.2
Rotations in three dimensions
We now begin our discussion of angular momentum using its geometric interpretation
as the generator of rotations in space. I should emphasize
Physics 6210/Spring 2007/Lecture 20
Lecture 20
Relevant sections in text: 2.6, 3.1
Gauge transformations (cont.)
Our proof that the spectrum of the Hamiltonian does not change when the potentials
are redened by a gauge transformation also indicates how we
Physics 6210/Spring 2007/Lecture 3
Lecture 3
Relevant sections in text: 1.2, 1.3
Spin states
We now model the states of the spin 1/2 particle. As before, we denote a state of the
particle in which the component of the spin vector S along the unit vector n
Physics 6210/Spring 2007/Lecture 4
Lecture 4
Relevant sections in text: 1.2, 1.3, 1.4
The spin operators
Finally we can discuss the denition of the spin observables for a spin 1/2 system. We
will do this by giving the expansion of the operators in a parti
Physics 6210/Spring 2007/Lecture 5
Lecture 5
Relevant sections in text: 1.21.4
An alternate third postulate
Here is an equivalent statement of the third postulate.*
Alternate Third Postulate
Let A be a Hermitian operator with an ON basis of eigenvectors |
Physics 6210/Spring 2007/Lecture 7
Lecture 7
Relevant sections in text: 1.6
Observables with continuous and/or unbounded values
We are now ready to turn to the quantum mechanical description of a (non-relativistic)
particle. We shall dene a (spinless) par
Physics 6210/Spring 2007/Lecture 8
Lecture 8
Relevant sections in text: 1.6
Momentum
How shall we view momentum in quantum mechanics? Should it be mass times velocity, or what? Our approach to the denition of momentum in quantum mechanics
will rely on a r
Physics 6210/Spring 2007/Lecture 9
Lecture 9
Relevant sections in text: 1.6, 1.7
Momentum wave functions
We have already indicated that one can use any continuous observable to dene a class
of wave functions. We have used the position observable to this e
Physics 6210/Spring 2007/Lecture 10
Lecture 10
Relevant sections in text: 1.7
Gaussian state
Here we consider the important example of a Gaussian state for a particle moving in
1-d. Our treatment is virtually identical to that in the text, but this exampl
Physics 6210/Spring 2007/Lecture 11
Lecture 11
Relevant sections in text: 1.7, 2.1
Product observables
We have seen how to build the Hilbert space for a composite system via the tensor
product construction. Let us now see how to build the observables. Let
Physics 6210/Spring 2007/Lecture 12
Lecture 12
Relevant sections in text: 2.1
The Hamiltonian and the Schrdinger equation
o
Consider time evolution from t to t + . We have
U (t + , t) = I +
i
H (t) + O( 2 ).
h
As usual, the unitarity of U implies that H
Physics 6210/Spring 2007/Lecture 13
Lecture 13
Relevant sections in text: 2.1
Example: Spin 1/2 in a uniform magnetic eld.
Let us consider the dynamical evolution of an electronic spin in a (uniform) magnetic
eld. We ignore the translational degrees of fr