PH 401 Dr. Cecilia Vogel

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

Review Schrödinger's Cat  _thought_experiment _thought_experiment

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 = S z  eigenvalues:  spin-up (+  /2)  spin-down (-  /2)  Initial state = superposition of spin-up and spin-down CAT SYSTEM  Operator = liveness  eigenvalues:  alive  dead  Initial state = superposition of alive and dead

Poor Cat is Dead  Initially (t=0), the electron is in superposition of spin- up and -down  Suppose we measure S z and find spin-down  at t= t1, the electron is no longer in superposition state  at t1, the state is 100% spin-down  Initially (t=0), the cat is in 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

There is Hope  At t1, the state is 100% spin-down  Now we measure S x, 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!

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 S z, and find either  spin-up, and quit  or spin-down, in which case we repeat the S x 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

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*

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.

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.

EPR Paradox  So if we make an S z 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 

EPR Paradox  If we just have the simple system with simple S z 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 EPR PAL