PH 401 Dr. Cecilia Vogel

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

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 t he eigenvalues of the z-component are m s  where m s ranges from –s to s in integer steps  for electron, s=1/2, so m s =+½

“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,  s leads to contradiction.  Let’s begin by assuming that there is a physical angle of rotation,  s, corresponding to spin rotation  in the same way that  corresponds to orbital angular momentum.

Pf by contradiction  If  s corresponds to spin rotation  in the same way that  corresponds to orbital angular momentum, then  would hold true (like for orbital)  OK, then what is the value of the function at  =0?  e 0 =1  And what is the value of the function at  =2  ?   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!

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

Spin Simultaneous Eigenstates  Because  there exists a complete set of simultaneous eigenstates of S 2 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!

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.

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

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 m ℓ =0

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 L 2 =0  Just means that the state is one with ℓ=m ℓ =0

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

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  ?  ….

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!

Review Schrödinger's Cat  _thought_experiment _thought_experiment