PH 401

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# PH 401 - PowerPoint PPT Presentation

PH 401. Dr. Cecilia Vogel. Review. Commutators and Uncertainty Angular Momentum Radial momentum. Spherically Symmetric Hamiltonian H-atom for example Eigenstates of H, L z , L 2 Degeneracy. Outline. Spherically Symmetric Problem. Suppose the potential energy depends only on r,

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Presentation Transcript
PH 401

Dr. Cecilia Vogel

Review
• Commutators and Uncertainty
• Angular Momentum
• Spherically Symmetric Hamiltonian
• H-atom for example
• Eigenstates of H, Lz, L2
• Degeneracy

Outline

Spherically Symmetric Problem
• Suppose the potential energy
• depends only on r,
• not q or f.
• such as for hydrogen atom
• then the Hamiltonian looks like.

=

Spherically Symmetric Problem
• The Hamiltonian in spherical coordinates, if V is symmetric
• Written in terms of pr, L2 and Lz:

=

=

=

Commutation
• Components of angular momentum commute with r, pr, and L2.
• Therefore Lz and L2 commute with H
• We can (and will) find set of simultaneous eigenstates of
• L2, Lz, and H
• Let Y(r,q,f) = R(r)f(q)g(f)
• Let quatum numbers be n, ℓ, mℓ
Separation of variables
• In the TISE, Hy=Ey, where
• there is only one term with f or f derivatives
• So this part separates

=

Eigenstate of Lz
• In fact
• So eigenvalue eqn for Lz is
• Lz|nlm> = m|nlm>
• means

=

Eigenstate of L2
• Also, angular derivatives only show up in L2 term, which also separates:
• FYI solution is Spherical Harmonics
• Note that L2=Lx2+Ly2+Lz2
• so

y

y =

Eigenstate of H
• As r goes to infinity, V(r) goes to zero
• For hydrogen atom
• So solution is (polynomial*e-r/na.)
• lowest order in polynomial is ,
• highest order in polynomial is n-1,
• so  is a non-negative integer,
• n is a positive integer, &
Degeneracy of Eigenstates
• Consider n=5
• 4th excited state of H-atom
• What are possible values of ?
• For each , what are possible values of m?
• for each n & , how many different states are there? “subshell”
• for each n, how many different states are there? “shell”
• what is the degeneracy of 4th excited state?
Spin Quantum Number
• Actually there turns out to be twice as many H-atom states as we just described.
• Introduce another quantum number that can have two values
• spin can be up or down (+½ or -½)
• It is called spin, but experimentally the matter of the electron is not spinning.
• It is like a spinning charged object, though, in the sense that
• it acts like a magnet, affected by B-fields
• it contributes to the angular momentum, when determining conservation thereof.
Quantized Lz
• Solution
• Impose boundary condition
• g(2p)=g(0)
• requires that ml=integer
• Lz is quantized
• or Lx or Ly, but not all
Eigenstate of Lx, Ly, and Lz
• Recall
• [Ly,Lz]=iLx is not zero
• so there’s not complete set of simultaneous eigenstates of Ly and Lz
• What if Lx=0?
• OK, but then also simultaneous eigenstate of Lx, and
• [Lx,Lz]=-iLy is not zero
• unless Ly=0
Eigenstate of Lx, Ly, and Lz
• Similarly
• [Lx,Ly]=iLz is not zero
• unless Lz=0
• So we can have a simultaneous eigenstate of Lx, Ly, Lz, and L2
• if the eigenvalues are all zero
• |n 0 0> is an eigenstate of all components of L
Component and Magnitude
• The fact that
• is a consequence of the fact that
• a component of a vector ( )
• can’t be bigger than
• magnitude of the vector ( )