PH 401 Dr. Cecilia Vogel

Review Outline  Time dependent perturbations  approximations  perturbation symmetry  Sx, Sy, Sz eigenstates  spinors, matrix representation  states and operators as matrices  multiplying them

Time Dependence  We have looked at stationary states  where measurable quantities don’t change at all with time  We have looked at non-stationary states  which develop and change with time  due to the superposition of energy levels  What about a system in a stationary state  that changes due to some external interaction  called a perturbation?

Unperturbed System  Let’s assume that we have solved the TISE  for the UNperturbed syste  with UNperturbed Hamitonian H o.  H o |  n > = E n |  n >  If our system begins in state  |  >=  c n |  n >  where c n =amplitude for state |  n >  then at time t, it will evolve to  |  >=  c n e -iE n t/  |  n >  Note that the probability of any state |  n > is still |c n | 2. The unperturbed system will not change energy levels.

Perturbing the System  Now let’s add a perturbing potential that depends on time  for example, it may be something we turn on at time t that wasn’t there before, it may be something that oscillates with time, like a light wave, etc  The full Hamiltonian is then  H=H o +V pert (t).  (Note: what I call V pert (t), your text calls H 1 )

Perturbed System  If our system begins in state  |  >=  c n (0)|  n >  where c n (0)= initial amplitude for state |  n >  then at time t, it will evolve to  |  >=  c n (t)e -iE n t/  |  n >  Because the perturbation depends on time, the amplitudes also depend on time.  The probability of any state |  n > changes with time. The perturbed system can indeed change energy levels.  Let’s see how….

TDSE  The state of our system  |  >=  c n (t)e -iE n t/  |  n >  must obey the TDSE  not the TISE, because it is not a stationary state of the full Hamiltonian  Plug into TDSE:  H|  >=i  ∂|  >/∂t  (H o +V pert (t))  c n (t)e -iE n t/  |  n >= i  ∂/∂t  c n (t)e -iE n t/  |  n >   (E n +V pert (t)) c n (t)e -iE n t/  |  n >= i  [   -iE n /  )c n (t)e -iE n t/  |  n >+  c’ n (t) e -iE n t/  |  n >]  Sooo…   c n (t)e -iE n t/  V pert (t)|  n >= i   c’ n (t) e -iE n t/  |  n >

Diff Eqn for Amplitudes  Sooo…   c n (t)e -iE n t/  V pert (t)|  n >= i   c’ n (t) e -iE n t/  |  n >  One more step to give us numbers rather than states:  Take an overlap of this with some final state |  f >  we can do this for any and all possible |  f >, so we don’t lose generality.  =  i  c’ f (t) e -iE f t/  =  c n (t)e -iE n t/   c’ f (t) =(-i/  )  c n (t)e i(E f -E n )t/ 

Time Dependence of c(t)  Thus the amplitudes evolve according to  c’ f (t) =(-i/  )  c n (t)e i(E f -E n )t/   This is a differential equation relating the derivative of any of the cn’s to the values of all the (infinitely many) other cn’s. OMG   We will not solve this, except in an approximate way.  If we assume that the perturbation is weak and we don’t wait too long,  then the state does not change much from what it was initially.

1 st Order Approximation  If we assume that the state does not change much from what it was initially.  ci=1, all other cn=0  Thus the amplitudes evolve according to  c’ f (t) =(-i/  ) e i(E f -E i )t/   In this case, the derivative = computable ftn of t, that can in principle be integrate to get c(t).

Matrix Element  The amplitudes evolve according to  c’ f (t) =(-i/  ) e i(E f -E i )t/   The evolution depends on the quantity   which is called a matrix element of the perturbing potential.  The matrix element is important, because it tells us what states can and cannot be excited by the potential.  If the matrix element is zero, then  c’ f (t)=0, and c f (t) remains zero for all time  that state does not get excited by the potential

Symmetry and Matrix Element  Under what circumstances is  =0?  and thus incapable of exciting that state?  If V pert commutes with Q, then  |  f > and |  i > cannot have different eigenvalues of Q  and V pert cannot change the eigenvalue of Q  Proof:  Consider initial and final states with initial and final eigenvalues of Q = q i and q f  = since they commute.  q f = q i  So… either q i = q f, or =0

Symmetry Example  If V pert commutes with Q, then  |  f > and |  i > cannot have different eigenvalues of Q  and V pert cannot change the eigenvalue of Q  Example:  If V pert = V pert (y,z) is not a function of x.  then V pert commutes with p x.  and this perturbation cannot change the particle’s x- component of momentum.  Classical analogue: If V does not depend on x, then there is no force in the x-dir and px does not change.  Example:  If V pert = V pert (r,  ) is not a function of .  then V pert commutes with L z.  and this perturbation cannot change the particle’s z- component of angular momentum

PAL