1. PH 401 Dr. Cecilia Vogel approx of notes

2. Recall Hamiltonian  In 1-D  H=p 2 /2m + V(x)  but in 3-D p 2 =px 2 +py 2 +pz 2.  and V might be a ftn of x, y, z

3. 3-D TISE  3-D TISE   where Del-squared =

4. 3-D Free particle  For a free particle, V=0   There are many solutions with the same energy  we can distinguish them by their  eigenvalues of px, py, and pz  Can we find a complete set of stationary states that are also eigenstates of px, py, and pz?

5. Commutators  Let me digress to define commutators  The commutator of operators A and B  is written [A,B]  and is defined by [A,B]=AB-BA  Often the order that you apply the operators matters  then AB is not equal to BA  but sometimes AB=BA,  then [A,B]=0  and we say that A and B commute.

6. Commutator of x and p  For example consider x and p x  [x,p x ]=xp x -p x x  Let this act on an arbitrary state  (xp x -p x x)  = (xp x  -p x x  ) ==  So.. [x,p x ]  =i   …for all   thus [x,p x ]=i   they do not commute

7. Commutator of x and p  generally  [position component,same component of momentum]=i   they do not commute  but [position component,same component of momentum]=0  they do commute  and [position component,position component] =0  they commute  and [momentum component,momentum component] =0  they commute, too

8. Playing with Commutators  [x,p x ]=i   so xp x =p x x+ i   Consider [x 2,p x ]= xxp x –p x xx  Let’s rearrange the left to look like the right, so some stuff will cancel  we can replace xp x =p x x+ i   Consider [x 2,p x ]= x(p x x+ i  )–p x xx  = xp x x+ i  x–p x xx  replace again  = (p x x+ i  )x+ i  x–p x xx = p x xx+ i  x+ i  x–p x xx  Finally canceling, [x 2,p x ]= 2 i  x

9. PAL Monday week 7 Prove that [r 2,p x ]=2i  x

10. Simultaneous Eigenstates  Can one have simultaneous eigenstates of A and B?  If [A,B]=0  then there exists a complete set of simultaneous eigenstates of A and B  every state can be written as a linear combo of these simultaneous eigenstates  If [A,B]  =0  then  could be a simultaneous eigenstate, even if there isn’t a complete set