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PH 401

PH 401. Dr. Cecilia Vogel. Review. Prove the radial H-atom solution Spin evidence spin angular momentum. Spin spin angular momentum not really spinning simultaneous eigenstates and measurement Schrödinger's cat. Outline. Spin Angular Momentum.

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PH 401

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  1. PH 401 Dr. Cecilia Vogel

  2. Review • Prove the radial H-atom solution • Spin • evidence • spin angular momentum • Spin • spin angular momentum • not really spinning • simultaneous eigenstates and measurement • Schrödinger's cat Outline

  3. Spin Angular Momentum • Spin is like other forms of angular momentum, in the sense that • it acts like a magnet, affected by B-fields • it contributes to the angular momentum, when determining conservation thereof. • The eigenvalues of the magnitude of the vector are • for electron, s=1/2, so • And the eigenvalues of the z-component are ms where msranges from –s to s in integer steps • for electron, s=1/2, so ms =+½

  4. “Spinning” is an imperfect model • Spin is UNlike other forms of angular momentum, in the sense that nothing is physically spinning! • For one thing, the electron is a point particle; how can a point spin? • For another thing, assuming that there is a spin angle, fs leads to contradiction. • Let’s begin by assuming that there is a physical angle of rotation, fs, corresponding to spin rotation • in the same way that f corresponds to orbital angular momentum.

  5. Pf by contradiction • If fs corresponds to spin rotation • in the same way that f corresponds to orbital angular momentum, then • would hold true (like for orbital) • OK, then what is the value of the function at f=0? • e0=1 • And what is the value of the function at f=2p? • So, which is it? It’s the same point in space, but is the function 1 or -1? • Wavefunction should be single-valued • CONTRADICTION! Cross it out!

  6. Spin Commutators • Spin is like other forms of angular momentum, in one more way… • it obeys the same type of commutation relations. • and similarly for cyclic permutations of x, y, z • and • where i =x or y or z

  7. Spin Simultaneous Eigenstates • Because • there exists a complete set of simultaneous eigenstates of S2 and Sz, • with quantum numbers s and ms. • Because • (and similarly for cyclic permutations of x, y, z) • there are NO simultaneous eigenstates of two different components of spin of electron • If electron is in an eigenstate of Sz (ms=+1/2, for ex) • then Sz is certain, but • Sx and Sy are uncertain!

  8. Simultaneous Eigenstates Revisited • Recall • there exists a complete set of simultaneous eigenstates of two operators, only if they commute • so there is not a complete set of simultaneous eigenstates of different components of spin OR orbital angular momentum • But, just because there is not a complete set, does not mean there are none.

  9. Simultaneous Eigenstates Revisited • Recall • there exists a simultaneous eigenstate, |ab> of two operators, A and B, if • Is that possible for two components of spin? • suppose • using the commutation relation, • this means • which means |ab> is an eigenstate of Sz, with eigenvalue zero • For electron, Sz has eigenvalues +½ only. • CONTRADICTION again

  10. Simultaneous Eigenstates Revisited • Recall • there exists a simultaneous eigenstate, |ab> of two operators, A and B, if • Is that possible for two components of orbital angular momentum? • suppose • using the commutation relation, • this means • which means |ab> is an eigenstate of Lz, with eigenvalue zero • That’s cool. • Just means that the state is one with mℓ=0

  11. Simultaneous Eigenstate of Ang Mom components • In the previous slide, we showed that a simultaneous eigenstate of Lx and Ly could exist • so long as it was also an eigenstate of Lz • with Lz=0 • That means it’s a simultaneous eigenstate of Lz and Lx (and Ly) • thus • which means • which means |ab> is an eigenstate of Lx, Ly, and Lz, ALL with eigenvalue zero • That’s cool. Then L2=0 • Just means that the state is one with ℓ=mℓ=0

  12. Simultaneous Eigenstates • The punchline is • there are NO simultaneous eigenstates of two different components of spin of electron • but there are simultaneous eigenstates of two different components of orbital angular momentum of electron, • and those are the states with ℓ=mℓ=0

  13. Simultaneous Eigenstates & Measurement • Suppose an electron is in a superposition state of spin-up and spin-down • it has an uncertain Sz • Then we measure Sz and find Sz= - ½  • now Sz is no longer uncertain • the measurement collapsed the wavefunction into an eigenstate of what we were measuring. • Since Sz is certain, Sx and Sy are uncertain • but there is nothing to stop us from measuring Sx. • What happens if we measure Sx and find +½ ? • ….

  14. Simultaneous Eigenstates & Measurement • We measured Sz and found Sz= - ½  • Then we measured Sx and found Sx=+½ ? …. • So our electron has Sz= - ½  and Sx =+½ ? • NO – that would be a simultaneous eigenstate of Sx and Sz, which is impossible! • When we measured Sx, we collapsed the wavefunction again • it is not in the same state it was in • it no longer has Sz = - ½  • instead it has collapsed into an eigenstate of Sx • If we measure Sz now, we have no idea what we’ll find!

  15. Review Schrödinger's Cat • http://en.wikipedia.org/wiki/Schroedinger's_cat#The_thought_experiment

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