Quantum Mechanics and Atomic Physics

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Quantum Mechanics and Atomic Physics

• Lecture 14
• Angular Momentum Operators
• http//www.physics.rutgers.edu/ugrad/361
• Prof. Eva Halkiadakis

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Last time

• Solved for S.E. in 3D using particle in a box
  example
• Using separation of variables we found that the
  total wavefunction is
• Quantization of energy is
• And found that energy degeneracy increases with
  energy

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Spherical Coordinates

• One of our goals is to solve S.E. in 3D for the
  Coulomb potential (1/r) which describes the
  hydrogen atom (next week)
• To do this we need to use spherical coordinates

Reed Chapter 6
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Angular Momentum

• Recall that angular momentum is
• It can be easily shown that in Cartesian
  coordinates

Note No x appears in Lx, No y appears in
Ly, And no z appears in Lz.
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Commutators of x,y,z components of L

• For example

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What does this mean?

• The components of L do not commute with each
  other!
• No simultaneous eigenstates!
• If you measure Lx ? get a certain value
• Next, measure Ly ? get a certain value
• Measure Lx again ? in general, you wont get the
  same value as before!
• We will revisit this in Chapter 8

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L operator in spherical coordinates
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Now the x,y,z components in spherical coordinates

• First the z component

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• Similarly,
• I leave it for you as an exercise to derive these
• Later, we will need to use the expression for Lz

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What about important quantity Lop2 ?
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Lets relate this to the Hamiltonian
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Central Potentials V( r )

• The S.E. for a particle of mass ? moving in a
  central potential V( r)
• Ive replaced m with ???to not be confused with
  new variable I will introduce shortly ml, and in
  reality it should be expressed as the reduced
  mass ??
• Again, lets use separation of variables, like we
  did last time

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• Some things to note about this equation

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Separation of variables

• (we will come back to this equation)
• Lets call the constant

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• Lets rewrite the RHS

So, full wavefunction is ?(r,???) R(
r)????????
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• Lets call the constant (-ml2)
• We will discover later that this is called the
  mangetic quantum number

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Solutions to ????
You will see later that we dont need the e-im?
term
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• Single-valuedness requires that
• We find that the mangnetic quantum number is
  quantized!

? is cyclic
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Lets go back to separated S.E.

• RHS depends on ?
• LHS depends on r
• Equal to a constant
• Lets call it l(l1) . Which is also equal to
  what weve been calling ?!

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• Solutions to ????
• They are associated Legendre Polynomials (see
  Chapter 6 for full expression)
• A few of them are
• Note special cases of ml0 are called Regular
  Legendre Polynomials

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Spherical Harmonics

• We usually deal with a combined angular
  dependence
• Y(?????????????
• These are called Spherical Harmonics (see Chapter
  6 for full expression)
• And is l called the orbital angular momentum
  quantum number

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Radial Solutions

• Solutions to R( r), depend on the potential
  energy
• We will revisit this next week

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Lets go back to

• Recall

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Total angular momentum is quantized!
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What about Lz?

• So, solutions exist only if l is an integer and l
  ml

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Space Quantization

• The magnetic quantum number ml expresses the
  quantization of direction of L
• So L can assume only certain angles, given above,
  with respect to the z-axis.
• This is called space quantization.

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Example

• For a particle with l2, what are the possible
  angels that L can make with the z-axis?

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Plot of Spherical harmonics

• 3D plots of
• ? is vertically up
• Independent of ?
• So rotationally symmetric around z-axis
• We will see later these will manifest themselves
  as the probability distributions

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Summary/Announcements

• Next time Central Potentials
• Next homework due on Monday Oct 27.
• You will have a quiz next week on Wed!