1 / 15

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.

ricky
Download Presentation

PH 401

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related