Modern Quantum Mechanics by gawde akshay & Napolitano Jim & Sakurai J

Author:gawde, akshay & Napolitano, Jim & Sakurai, J. [gawde, akshay]
Language: eng
Format: epub
Publisher: Panda Books
Published: 2020-08-29T16:00:00+00:00

4.2 Discrete Symmetries, Parity, or Space Inversion

273

Not all wave functions of physical interest have definite parities in the sense of (4.2.24). Consider, for instance, the momentum eigenket. The momentum operator anticommutes with the parity operator, so the momentum eigenket is not expected to be a parity eigenket. Indeed, it is easy to see that the plane wave, which is the wave function for a momentum eigenket, does not satisfy (4.2.24).

An eigenket of orbital angular momentum is expected to be a parity eigenket

because L and π commute [see (4.2.11)]. To see how an eigenket of L 2 and Lz behaves under parity, let us examine the properties of its wave function under space inversion,

x | α , lm = Rα ( r ) Y m

l ( θ , φ ).

(4.2.25)

The transformation x → −x is accomplished by letting

r → r

θ → π − θ

(cos θ → − cos θ )

(4.2.26)

φ → φ + π

( eimφ → (−1) meimφ ).

Using the explicit form of

(2 l + 1)( l − m )!

Y m =

Pm

l

(−1) m

4 π ( l + m )!

l (cos θ ) eimφ

(4.2.27)

for positive m , with (3.6.38), where

l −| m |

| m |

( l + | m |)!

d

P

(cos θ ) = (−1) m + l

sin−| m | θ

sin2 l θ ,

(4.2.28)

l

2 ll !

( l − | m |)!

d (cos θ )

we can readily show that

Y m →

l

(−1) lY m

l

(4.2.29)

as θ and φ are changed, as in (4.2.26). Therefore, we can conclude that π | α , lm = (−1) l | α , lm .

(4.2.30)

It is actually not necessary to look at Y m ; an easier way to obtain the same result l

is to work with m = 0 and note that Lr ±| l , m = 0( r = 0,1, ... , l ) must have the same parity because π and ( L ±) r commute.

Let us now look at the parity properties of energy eigenstates. We begin by

stating a very important theorem.

Theorem 4.1.

Suppose

[ H , π ] = 0

(4.2.31)

and | n is a nondegenerate eigenket of H with eigenvalue En :

H | n = En | n ;

(4.2.32)

then | n is also a parity eigenket.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.