QMGreensite_merged

8.4. THEFINITESQUAREWELL:BOUNDSTATES 137

substituteinto(8.74)

Ae−ika+Beika =

[

1

2

(1+

k

q

)ei(k−^2 q)a+

1

2

(1−

k

q

)ei(k+2q)a

]

E

k(Ae−ika−Beika) = q

[

1

2

(1+

k

q

)ei(k−^2 q)a−

1

2

(1−

k

q

)ei(k+2q)a

]

E (8.76)

SolvingforEintermsofA,wefind

2 kAe−ika =

[

(k+q)

1

2

(1+

k

q

)ei(k−^2 q)a+(k−q)

1

2

(1−

k

q

)ei(k+2q)a

]

E

=

[

(k+

k^2

2 q

+

q

2

)ei(k−^2 q)a+(k−

k^2

2 q

−

q

2

)ei(k+2q)a

]

E (8.77)

sothat

E

A

=

2 e−ika

(1+ 2 kq+ 2 qk)ei(k−^2 q)a+(1− 2 kq− 2 qk)ei(k+2q)a

=

e−^2 ika

cos(2qa)+ 21 ( 2 kq+ 2 qk)(e−^2 iqa−e^2 iqa)

=

e−^2 ika

cos(2qa)− 2 i(kq+qk)sin(2qa)

(8.78)

Inasimilarway,onecansolveforBintermsofA

B

A

=

1

2 ie

− 2 ika(q

k−

k

q)sin(2qa)

cos(2qa)−^12 i(qk+kq)sin(2qa)

(8.79)

Fromtheseratios,weobtainthetransmissionandreflectioncoefficients

T =

∣∣

∣∣E

A

∣∣

∣∣

2

=

1

cos^2 (2qa)+^14 (kq+qk)^2 sin^2 (2qa)

R =

∣∣

∣∣B

A

∣∣

∣∣

2

=

1

4 (

q

k−

k

q)

(^2) sin^2 (2qa)
cos^2 (2qa)+^14 (kq+qk)^2 sin^2 (2qa)

(8.80)

anditiseasytocheckthat
R+T= 1 (8.81)

asrequiredbyconservationofprobability.
NownoticethatsomethinginterestinghappenswhentheenergyEoftheincoming
particleissuchthat

2 qa= 2

√

2 m(E+V 0 )

a

h ̄

=nπ (8.82)