Page 123 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
′
—ω bb b ψ =
o
which gives with (3.55) + o E bψ o (3.56)
+
′
(
—ω 1 + b b) bψ = E bψ o (3.57)
o
o
or
′
+
—ω b bbψ = ( E – —ω)bψ (3.58)
o o o
′
(3.58) means that there is a ground state ψ = bψ with an even smaller
o o
energy than E ′ . This is contradictory to the presupposition that ψ be
o o
the ground state. Even if the first choice of ψ had not been the right
o
ground state, repeated execution of the operation in (3.56) would lead us
′
to an energy eigenvalue E → – ∞ . This cannot be, since the potential
o
energy is bound, with its minimum at U = 0 . On the other hand, (3.56)
is a valid equation with a valid operation. Thus the only solution is that
bψ = 0 . Inserting the definition of the operator b given in (3.53)
o
yields
 d 
ξ
 ------ +  ψ ξ() = 0 (3.59)
o
 dξ 
with the solution

1  ξ 2 
Ground–State ψ ξ() = --------------- exp⋅  – ----- (3.60)
2 
⁄
o
14
π ()
Oscillator
Wavefunction
bψ = 0 implies that the l.h.s. of (3.54) is zero for the ground–state
o
′
wavefunction and thus we have E = 0 . This gives us the ground state
o
⁄
energy E = ( —ω 2) . We perform the same operation as in (3.56), now
o
letting b + act on the Schrödinger equation for the ground state, which
gives
+ + +
′
—ω b b b ψ = E b ψ (3.61)
o o o
120 Semiconductors for Micro and Nanosystem Technology