Download Chapter 4 Free and Confined Electrons

ECE 685 Nanoelectronics
Chapter 4
Free and Confined Electrons
Lecture given by Qiliang Li
Dept. of Electrical and Computer Engineering
George Mason University
§ 4.5.2 Parabolic Well – Harmonic Oscillator
2
d
F  ma  m 2 x   Kx
dt
x  sin( t K m )
w K m
1 2 1 2
2
V ( x )  Kx  w0 me x
2
2
§ 4.5.2 Parabolic Well – Harmonic Oscillator
The Schrodinger’s equation is:
h2 d 2 1 2 2
(
 w0 mx )( x )  E( x )
2
2m dx
2
§ 4.5.2 Parabolic Well – Harmonic Oscillator
divergent
Let
§ 4.5.2 Parabolic Well – Harmonic Oscillator
Let
§ 4.5.2 Parabolic Well – Harmonic Oscillator
∞
𝐻 𝜌 ~
𝑛=0
1 𝑛
𝑛 𝜌
!
2
∞
~
𝑚=0
1 2𝑚
𝜌
~exp⁡
(𝜌2 )
𝑚 !
Therefore, H(x) grows like exp(x^2), producing unphysical
diverging solution.
So the coefficients beyond a given n should vanish, the
infinite series becomes a finite polynomial. So we should
have: 𝐴
2𝜖 − 2𝑛 − 1 = 0
=0
𝑛+2
§ 4.5.2 Parabolic Well – Harmonic Oscillator
n is a non-negative integer: 0, 1, 2, …
§ 4.5.2 Parabolic Well – Harmonic Oscillator
ladder operator
1
𝐻Ψ 𝑥 = 𝐸Ψ 𝑥 = (𝑛 + )ℏ𝑤Ψ 𝑥
2
𝐻= 𝑁+
1
1
ℏ𝑤 = (𝑎+ 𝑎 + )ℏ𝑤
2
2
𝑎 𝜓𝑛 >= 𝑛 𝜓𝑛−1 >
𝑎 + 𝜓𝑛 >= 𝑛 + 1 𝜓𝑛+1 >
𝑎+ 𝑎 𝜓𝑛 >= 𝑛 𝜓𝑛 >
𝑎𝑎+ 𝜓𝑛 >= (𝑛 + 1) 𝜓𝑛 >
ℏ2 𝑑2
1 2 2
𝑃2 1 2 2
𝐻=−
+ 𝑤 𝑚𝑥 =
+ 𝑤 𝑚𝑥
2𝑚 𝑑𝑥 2 2
2𝑚 2
1
𝑃2 1 2 2
𝑎 𝑎 + ℏ𝑤 =
+ 𝑤 𝑚𝑥
2
2𝑚 2
+
𝑎=
𝑚𝑤
𝑖
(𝑥 +
𝑃)
2ℏ
𝑚𝑤
𝑎+ =
𝑚𝑤
𝑖
(𝑥 −
𝑃)
2ℏ
𝑚𝑤
§ 4.5.2 Parabolic Well – Harmonic Oscillator
Use ladder operator to find the wave function:
𝑎|𝜓0 >= 0
𝑥𝜓0 𝑥 +
ℏ 𝑑
𝜓 𝑥 =0
𝑚𝑤 𝑑𝑥 0
ln 𝜓0 𝑥
𝑚𝑤
𝜓0 𝑥 =
𝜋ℏ
=−
1
4
𝑚𝑤 2
𝑥
2ℏ
exp⁡
(−
𝑚𝑤 2
𝑥 )
2ℏ
𝑎|𝜓1 >= |𝜓0 >
𝑚𝑤
𝑚𝑤 ℏ 𝑑
𝑚𝑤
𝑥𝜓1 𝑥 +
𝜓1 𝑥 =
2ℏ
2ℏ 𝑚𝑤 𝑑𝑥
𝜋ℏ
1
4
exp⁡
(−
𝑚𝑤 2
𝑥 )
2ℏ
§ 4.5.2 Parabolic Well – Harmonic Oscillator
∞
Let:
𝑚𝑤 2
𝑎𝑛 𝑥 𝑛 𝑒 − 2ℏ 𝑥
𝜓1 𝑥 =
𝑛=0
𝑚𝑤
2ℏ
∞
𝑚𝑤 2
−
𝑥
𝑎𝑛 𝑥 𝑛+1 𝑒 2ℏ
𝑛=0
=
𝑚𝑤
𝜋ℏ
1
4
𝑚𝑤 ℏ
+
2ℏ 𝑚𝑤
exp⁡
(−
Only a1 is not 0:
∞
𝑚𝑤 2
−
𝑥
𝑛𝑎𝑛 𝑥 𝑛−1 𝑒 2ℏ
𝑛=0
𝑚𝑤 ℏ
+
2ℏ 𝑚𝑤
𝑚𝑤 2
𝑥 )
2ℏ
𝑚𝑤 ℏ
𝑚𝑤
𝑎1 =
2ℏ 𝑚𝑤
𝜋ℏ
𝑚𝑤
𝑎1 =
𝜋ℏ
𝑚𝑤
𝜓1 𝑥 =
𝜋ℏ
1
4
1
4
1
4
2𝑚𝑤
ℏ
𝑚𝑤 2
2𝑚𝑤
𝑥 𝑒 − 2ℏ 𝑥
ℏ
Similarly, we can find more wavefunction…
∞
𝑎𝑛 𝑥 𝑛 −
𝑛=0
𝑚𝑤 2
𝑚𝑤
−
𝑥
𝑥 𝑒 2ℏ
ℏ
§ 4.5.2 Parabolic Well – Harmonic Oscillator
In (A-B), the particle (represented
as a ball attached to a spring)
oscillates back and forth. In (C-H),
some solutions to the
Schrödinger Equation are shown,
where the horizontal axis is
position, and the vertical axis is
the real part (blue) or imaginary
part (red) of the wavefunction.
(C,D,E,F), but not (G,H), are
energy eigenstates. (H) is a
coherent state, a quantum state
which approximates the classical
trajectory.
§ 4.5.3 Triangular Well
§ 4.5.3 Triangular Well
ℏ2 𝑑2
(−
+ 𝑐𝑥)Ψ 𝑥 = 𝐸Ψ 𝑥
2𝑚 𝑑𝑥 2
2𝑚𝑐
𝜉=
ℏ2
1
3
𝐸
(𝑥 − )
𝑐
𝑑2
𝜓 𝜉 − 𝜉𝜓 𝜉 = 0
𝑑𝜉 2
§ 4.5.3 Triangular Well
Example:
§ 4.6 Electron confined to atom
See lecture note