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PH 401

PH 401. Dr. Cecilia Vogel. Review. Spin spin angular momentum not really spinning simultaneous eigenstates and measurement Schrödinger's cat. Resuscitating Schrödinger's cat Pauli Exclusion Principle EPR Paradox. Outline. Review Schrödinger's Cat.

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PH 401

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  1. PH 401 Dr. Cecilia Vogel

  2. Review • Spin • spin angular momentum • not really spinning • simultaneous eigenstates and measurement • Schrödinger's cat • Resuscitating Schrödinger's cat • Pauli Exclusion Principle • EPR Paradox Outline

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

  4. Set Up Analogue • Let’s pretend that a cat could really be considered a quantum system. Its liveness is a two-eigenvalue system, like an electron’s spin. • Let’s draw an analogy (that should not be taken seriously!) ELECTRON SYSTEM • Operator = Sz • eigenvalues: • spin-up (+/2) • spin-down (-/2) • Initial state = 50-50 superposition of spin-up and spin-down CAT SYSTEM • Operator = liveness • eigenvalues: • alive • dead • Initial state = 50-50 superposition of alive and dead

  5. Poor Cat is Dead • Initially (t=0), the cat is in 50-50 superposition of alive and dead • Suppose we measure liveness and find the cat is dead • at t= t1, the cat is no longer in superposition state • at t1, the state is 100% dead • Initially (t=0), the electron is in 50-50 superposition of spin-up and -down • Suppose we measure Sz and find spin-down • at t= t1, the electron is no longer in superposition state • at t1, the state is 100% spin-down

  6. There is Hope • At t1, the state is 100% spin-down • Now we measure Sx, finding spin-right (or left) • at t= t2, the electron is no longer in spin-down state • it is in a spin-right (or left) state • which is a superposition of spin-up and spin-down • At t1, the state is 100% dead  • Now we measure B, an operator incompatible with liveness • that places the cat in an eigenstate of B • It can’t at the same time be in an eigenstate of liveness, • so its state is a superposition of alive and dead!

  7. Resuscitating the Cat • At t= t2, the electron is back in a superposition of spin-up and spin-down (as it was at t=0) • Again we measure Sz, and find either • spin-up, and quit • or spin-down, in which case we repeat the Sx measurement and go thru the loop repeatedly until we get spin-up • At t= t2, the cat is back in a superposition of alive and dead (as it was at t=0) • Again we measure liveness, and find either • ALIVE, yay  • or dead, in which case we repeat the B measurement and go thru the loop repeatedly until we get a cat that is alive

  8. Good Luck • The ability to resuscitate the cat • hinges on determining an operator B • that is incompatible with the liveness operator • [liveness, B] ≠ 0 • I have no idea how you would express “liveness” as a quantum operator, • much less how you would find a measurable quantity, B, that is incompatible with it. • Classical observables do not behave this way – incompatibility applies to the quantum realm only. • Why? *shrugs*

  9. Pauli Exclusion Principle • In chemistry we learn that no two electrons in an atom can have the same set of quantum numbers. • The Pauli Exclusion Principle is even more general: • No two identical fermions can occupy the same quantum state. • What is a fermion? Any particle with half-integer spin, like electrons, protons, neutrons, muons,… • in contrast to bosons, which have integer spin, like the photon, gluon, … • All electrons are identical – you can’t tell one from another.

  10. Quantum Entanglement • The Pauli Exclusion Principle gives us one way to create entangled particles. • Suppose we have two electrons in the ground state of a He atom. • They cannot be in the same state, so if one is spin-up, the other is spin-down, • even if both are in states of superposition of spin-up and spin-down. • These electrons are entangled, because a measurement of the spin-state of one determines the spin-state of the other. • If one is found to be spin-up, the other must be spin-down.

  11. EPR Paradox • So if we make an Sz measurement on one of those electrons, we collapse its wavefunction. • If we find spin-up, then its state is no longer a superposition, but rather 100% spin-up. • ALSO… the other electron is no longer in a superposition state, but rather 100% spin-down. • We have collapsed its wavefunction without interacting directly with it! • This violates separable local realism… • Either the two electrons cannot be treated separately • or there is an instantaneous interaction between them, which violates causality in relativity • or reality doesn’t exist 

  12. EPR Paradox • If we just have the simple system with simple Sz measurements described above, • then we can’t prove that the electrons were ever in a superposition state • maybe they were just 100% spin-up and 100% spin down all along • like classical statistics. • We need a more complicated system to prove that this doesn’t work, and that is our PAL • If you want to read more, look up EPR or Bell’s inequality (the inequality that fails in the PAL) • EPR PAL

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