7. Angular Momentum

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7. Angular Momentum
7A. Angular Momentum Commutation
Why it doesnt commute

• The order in which you rotate things makes a
  difference, ?1?2 ? ?2?1
• We can use this to work out commutation relations
  for the Ls
• It can be done more easily directly
• Recall
• Also recall
• We will calculate the following to second order
  in ?
• If rotations commuted, both sides would be the
  identity relation

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Calculating the Left Side

• To second order in ?
• Second half is same thing with ? ? ?

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Calculating the Right Side

• To second order in ?
• Second half is same thing with ? ? ?

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Matching the Two Sides

• To second order in ?
• Now match the two sides

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Levi-Civita Symbol

• Generalizing, we have
• Define the Levi-Civita symbol
• Then we write
• We will call any three Hermitian operators Jthat
  work this way generalized angular momentum

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7B. Generalized Angular Momentum
J2 and the raising/lowering operators

• What can we conclude just from the commutation
  relations?
• If J commutes with the Hamiltonian, than we
  cansimultaneously diagonalize H and one
  component of J
• Normally pick Jz
• Define some new operators
• Reverse these if we want
• These satisfy the following properties
• Proof by homework problem

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Eigenstates

• Since J2 commutes with Jz, we can diagonalize
  them simultaneously
• We will (arbitrarily for now) choose an odd way
  to write the eigenvalues
• Note that j and m are dimensionless
• Note that J2 has positive eigenvalues
• We can choose j to be non-negative
• We can let J? act on any state j,m? to produce a
  new state J? j,m?
• This new state must be proportional to

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Eigenstates (2)

• To find proportionality, consider
• This expression must not be negative
• When it is positive, then we have
• Choose the phase positive
• Conclusion given a state j,m?, we can produce a
  series of other states
• Problem if you raise or lower enough times, you
  eventually get m gt j
• Resolution You must have

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Summary

• Eigenstates look like
• The values of m are
• There are 2j 1 of them
• Since 2j1 is an integer
• We can use these expressions to write out Js as
  matrices of size (2j 1) ? (2j 1)
• First, pick an order for your eigenstates,
  traditionally
• The matrix Jz is trivial to write down, and is
  diagonal
• The matrix J is a little harder, and is just
  above the diagonal
• You then get J- J and can find Jx and Jy

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Sample Problem

• Basis states
• Jz is diagonal
• J just above the diagonal

Write out the matrix form for J for j 1
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Sample Problem (2)
Write out the matrix form for J for j 1

• Now work out Jx and Jy
• As a check, find J2

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Special Cases and Pauli Matrices

• The matrices for j 0 are really simple
• We sometimes call this the scalar representation
• The j ½ is called the spinor representation,
  and is important
• There are only two states
• Often these states abbreviated
• The corresponding 2?2 matrices are written in
  terms of the Pauli matrices
• The Pauli matrices are given by
• Useful formulas

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7C. Spherically Symmetric Problems
Spherical Coordinates

• Consider this Hamiltonian
• All components of L commute with H, because they
  commute with R2
• It makes sense to choose eigenstates of H, L2 and
  Lz
• It seems sensible to switch to spherical
  coordinates
• We write Schrödingers equation in spherical
  coordinates

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L in Spherical coordinates (1)

• We need to write L in spherical coordinates
• Start by writing angular derivatives out
• Its not hard to get Lz from these equations

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L in Spherical coordinates (2)

• Now its time to get clever consider
• And we get clever once more

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Other Operators in Spherical coordinates

• It will help to get the raising and lowering
  operators
• And we need L2
• Compare to Schrödinger

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Solving Spherically Symmetric Problems

• Rewrite Schrödingers equation
• Our eigenstates will be
• The angular properties are governed by l and m
• This suggests factoring ? into angular and radial
  parts
• Substitute into Schrodinger
• Cancel Y

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The Problem Divided

• It remains to find and normalize R and Y
• Note that Y problem is independent of the
  potential V
• Note that the radial problem is a 1D problem
• Easily solved numerically
• Normalization
• Split this up how you want, but normally

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7D. The Spherical Harmonics
Dependence on ?, and m restrictions

• We will call our angular functions spherical
  harmonics and label them
• We previously found
• For general angular momentum we know
• We can easily determine the ? dependence of the
  spherical harmonics
• Also, recall that ? 0 is the same as ? 2?
• It follows that m (and therefore l) is an integer

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Finding one Spherical Harmonic

• We previously found
• For general angular momentum
• If we lower m l, we must get zero
• Normalize it

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Finding All Spherical Harmonics

• To get the others, just raise this repeatedly
• Sane people, or those who wish to remain so, do
  not use this formula
• Many sources list them
• P. 124 for l 0 to 4
• Computer programs can calculate them for you
• Hydrogen on my website

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Properties of Spherical Harmonics

• They are eigenstates of L2 and Lz
• They are orthonormal
• They are complete any angular function can be
  written in terms of them
• This helps us write the completion relation

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More Properties of Spherical Harmonics

• Recall parity commutes with L
• It follows that
• Hence when you let parity act, youmust be
  getting essentially the same state
• Recall L2 is real but Lz is pure imaginary
• Take the complex conjugate of our relations
  above
• This implies
• It works out to

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The Spherical Harmonics
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7E. The Hydrogen Atom
Changing Operators

• Hamiltonian for hydrogen (SI units)
• These operators have commutators
• Classically, what we do
• Total momentum is conserved
• Center of mass moves uniformly
• Work in terms of relative position
• Quantum mechanically Lets try
• Find commutation operators for these
• Proof by homework problem
• Find the new Hamiltonian
• Proof by homework problem

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Reducing the problem 6D to 3D

• Note that for actual hydrogen, ? is essentially
  the electron mass
• Split the Hamiltonian into two pieces
• These two pieces have nothing to do with each
  other
• It is essentially two problems
• Hcm is basically a free particle of mass M,
• It is trivial to solve
• The remaining problem is effectively a single
  particle of mass ? in a spherically symmetric
  potential

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Reducing the problem 3D to 1D

• Because the problem is spherically symmetric, we
  will have states
• These will have wave functions
• The radial wave function will satisfy
• Note that m does not appear in this equation, so
  R wont depend on it
• We will focus on bound states E lt 0

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The Radial Equation For r Large and Small

• Lets try to approximate behavior at r 0 and r
  ?
• Large r keep dominant terms, ignore those with
  negative powers of r
• Define a such that
• Then we have
• Want convergent
• Now, guess that for small r we have
• Substitute in, keeping smallest powers of r
• Want it convergent

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The Radial Equation Removing Asymptotic

• Factor out the expected asymptotic behavior
• This is just a definition of f(r)
• Substitute in, multiply by 2?/?2
• Define the Bohr radius

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The Radial Equation Taylor Expansion

• Write f as a power series around the origin
• Recall that at small r it goes like rl
• Substitute in
• Gather like powers of r
• On right side, replace i ? i 1
• On left side, first term vanishes
• These must be identical expressions, so

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Are We Done?

• It looks like we have a solution for any E
• Pick fl to be anything
• Deduce the rest by recursion
• Now just normalize everything
• Problem No guarantee it is normalizable
• Study the behavior at large i
• Only way to avoid this catastrophe is to make
  sure some f vanishes, say fn

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Summarizing Everything

• Because the exponential beats the polynomial,
  these functions are now all normalizable
• Arbitrarily pick fl gt 0
• Online Hydrogen or p. 124
• Note that n gt l, n is positive integer
• Include the angular wave functions
• Restrictions on the quantum numbers
• Another way of writing the energy
• For an electron orbiting a nucleus, ? is almost
  exactly the electron mass, ?c2 511 keV

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Radial Wave Functions
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Sample Problem
What is the expectation value for ?R? for a
hydrogen atom in the state n,l,m? 4,2,-1??

• The spherical harmonics are orthonormal over
  angles

gt integrate(radial(4,2)2r3,r0..infinity)
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Degeneracy and Other Issues

• Note energy depends only on n, not l or m
• Not on m because states related by rotation
• Not on l is an accident accidental degeneracy
• How many states with the same energy En?
• 2l 1 values of m
• l runs from 0 to n 1
• Later we will learn about spin, and realize there
  are actually twice as many states
• Are our results truly exact?
• We did include nuclear recoil, the fact that the
  nucleus has finite mass
• Relativistic effects
• Small for hydrogen, can show v/c ? 1/137
• Finite nuclear size
• Nucleus is 104 to 105 times smaller
• Very small effect
• Nuclear magnetic field interacting with the
  electron

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7E. Hydrogen-Like Atoms
Other Nuclei

• Can we apply our formulas to any other systems?
• Other atoms if they have only one electron in
  them
• The charge on the nucleus multiplies potential by
  Z
• Reduced mass essentially still the electron mass
• Just replace e2 by e2Z
• Atom gets smaller
• But still much larger than the nucleus
• Relativistic effects get bigger
• Now v/c Z?

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Bizarre atoms

• We can replace the nucleus or the electron with
  other things
• Anti-muon plus electron
• Anti-muon has same charge as proton, and much
  more mass than electron
• Essentially identical with hydrogen
• Positronium anti-electron (positron) plus
  electron
• Same charge as proton
• Positrons mass electrons mass
• Reduced mass and energy states reduced by half
• Nucleus plus muon
• Muon 207 times heavier than electron
• Atom is 207 times smaller
• Even inside a complex atom, muon sees essentially
  bare nucleus
• Atom small enough that for large Z, muon is
  partly inside nucleus
• Anti-hydrogen anti-proton plus anti-electron
• Identical to hydrogen