PH 401 Dr. Cecilia Vogel approx of notes
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-D TISE 3-D TISE where Del-squared =
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?
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.
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
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
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
PAL Monday week 7 Prove that [r 2,p x ]=2i x
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