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PH 401

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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PH 401

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  1. PH 401 Dr. Cecilia Vogel

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

  3. 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. =

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

  5. 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

  6. Separation of variables • In the TISE, Hy=Ey, where • there is only one term with f or f derivatives • So this part separates =

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

  8. 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 =

  9. Eigenstate of H • Radial part of eqn • 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, &

  10. 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?

  11. 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.

  12. Quantized Lz • Solution • Impose boundary condition • g(2p)=g(0) • requires that ml=integer • Lz is quantized • or Lx or Ly, but not all

  13. 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

  14. 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

  15. 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 ( )

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