p40_027.pdf

27. The energy levels are given by

E n x ,n y = h 2

8 m

n x

L x

+ n y

L y

h 2

8 mL 2

n x + n y

where the substitutions L x = L and L y =2 L were made. In units of h 2 / 8 mL 2 , the energy levels are

given by n x + n y / 4. The lowest ﬁve levels are E 1 , 1 =1 . 25, E 1 , 2 =2 . 00, E 1 , 3 =3 . 25, E 2 , 1 =4 . 25, and

E 2 , 2 = E 1 , 4 =5 . 00. It is clear that there are no other possible values for the energy less than 5. The

frequency of the light emitted or absorbed when the electron goes from an initial state i to a ﬁnal state

f is f =( E f −

−→

1,1), 1 . 25(1,3

−→

1,2), 2 . 25(2,1

−→

1,2), 3 . 00 (2,2

−→

1,2), 1 . 00 (2,1

−→

1,3), 1 . 75(2,2

−→

1,3),

0 . 75(2,2

−→

2,1), all in units of h/ 8 mL 2 .

E i ) /h , and in units of h/ 8 mL 2 is simply the diﬀerence in the values of n x + n y / 4for

the two states. The possible frequencies are 0 . 75(1,2 −→ 1,1), 2 . 00 (1,3 −→ 1,1), 3 . 00 (2,1 −→ 1,1), 3 . 75

(2,2