Chapter 4 Free and Confined Electrons

Lecture given by Qiliang LiDept. of Electrical and Computer Engineering

George Mason University

ECE 685 Nanoelectronics

§ 4.5.2 Parabolic Well – Harmonic Oscillator

220

2

2

1

2

1)( xmwKxxV eKxx

dt

dmmaF

2

2

)sin( mKtx

mKw

§ 4.5.2 Parabolic Well – Harmonic Oscillator

The Schrodinger’s equation is:

)()()2

1

2( 22

02

22

xExmxwdx

d

m

h

§ 4.5.2 Parabolic Well – Harmonic Oscillator

Let

divergent

§ 4.5.2 Parabolic Well – Harmonic Oscillator

Let

§ 4.5.2 Parabolic Well – Harmonic Oscillator

𝐻ሺ𝜌ሻ~ 1ቀ𝑛2ቁ!𝜌𝑛 ∞

𝑛=0 ~ 1ሺ𝑚ሻ!𝜌2𝑚 ∞

𝑚=0 ~exp (𝜌2) 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: 𝐴𝑛+2 = 0 2𝜖− 2𝑛− 1 = 0

§ 4.5.2 Parabolic Well – Harmonic Oscillator

n is a non-negative integer: 0, 1, 2, …

§ 4.5.2 Parabolic Well – Harmonic Oscillatorladder operator 𝐻Ψሺ𝑥ሻ= 𝐸Ψሺ𝑥ሻ= (𝑛+ 12)ℏ𝑤Ψሺ𝑥ሻ

𝐻= ൬𝑁 + 12൰ℏ𝑤= (𝑎+𝑎+ 12)ℏ𝑤

𝑎ห𝜓𝑛 >= ξ𝑛ห𝜓𝑛−1 >

𝑎+ห𝜓𝑛 >= ξ𝑛+ 1ห𝜓𝑛+1 >

𝑎+𝑎ȁ"𝜓𝑛 >= 𝑛ȁ"𝜓𝑛 >

𝑎𝑎+ȁ"𝜓𝑛 >= (𝑛+ 1)ȁ"𝜓𝑛 >

𝐻= − ℏ22𝑚 𝑑2𝑑𝑥2 + 12𝑤2𝑚𝑥2 = 𝑃22𝑚+ 12𝑤2𝑚𝑥2

൬𝑎+𝑎+ 12൰ℏ𝑤= 𝑃22𝑚+ 12𝑤2𝑚𝑥2

𝑎 =ට𝑚𝑤2ℏ (𝑥+ 𝑖𝑚𝑤𝑃)

𝑎+ = ට𝑚𝑤2ℏ (𝑥− 𝑖𝑚𝑤𝑃)

§ 4.5.2 Parabolic Well – Harmonic OscillatorUse ladder operator to find the wave function:

𝑎|𝜓0 >= 0

𝑥𝜓0ሺ𝑥ሻ+ ℏ𝑚𝑤 𝑑𝑑𝑥𝜓0ሺ𝑥ሻ= 0

ln൫𝜓0ሺ𝑥ሻ൯= −𝑚𝑤2ℏ 𝑥2

𝜓0ሺ𝑥ሻ= ቀ𝑚𝑤𝜋ℏቁ14 exp (−𝑚𝑤2ℏ 𝑥2)

𝑎|𝜓1 >= |𝜓0 >

ට𝑚𝑤2ℏ 𝑥𝜓1ሺ𝑥ሻ+ට

𝑚𝑤2ℏ ℏ𝑚𝑤 𝑑𝑑𝑥𝜓1ሺ𝑥ሻ= ቀ𝑚𝑤𝜋ℏቁ14 exp (−𝑚𝑤2ℏ 𝑥2)

𝜓1ሺ𝑥ሻ= 𝑎𝑛𝑥𝑛∞𝑛=0 𝑒−𝑚𝑤2ℏ 𝑥2

ට𝑚𝑤2ℏ 𝑎𝑛𝑥𝑛+1∞

𝑛=0 𝑒−𝑚𝑤2ℏ 𝑥2 +ට𝑚𝑤2ℏ ℏ𝑚𝑤 𝑛𝑎𝑛𝑥𝑛−1∞

𝑛=0 𝑒−𝑚𝑤2ℏ 𝑥2 +ට𝑚𝑤2ℏ ℏ𝑚𝑤 𝑎𝑛𝑥𝑛∞

𝑛=0 ቀ−𝑚𝑤ℏ 𝑥ቁ𝑒−𝑚𝑤2ℏ 𝑥2

= ቀ𝑚𝑤𝜋ℏቁ14 exp (−𝑚𝑤2ℏ 𝑥2)

ට𝑚𝑤2ℏ ℏ𝑚𝑤𝑎1 = ቀ

𝑚𝑤𝜋ℏቁ14

𝑎1 = ቀ𝑚𝑤𝜋ℏቁ14ඨ2𝑚𝑤ℏ

𝜓1ሺ𝑥ሻ= ቀ𝑚𝑤𝜋ℏቁ14ඨ2𝑚𝑤ℏ 𝑥 𝑒−𝑚𝑤2ℏ 𝑥2

§ 4.5.2 Parabolic Well – Harmonic Oscillator

Let:

Only a1 is not 0:

Similarly, we can find more wavefunction…

§ 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(− ℏ22𝑚 𝑑2𝑑𝑥2 + 𝑐𝑥)Ψሺ𝑥ሻ= 𝐸Ψሺ𝑥ሻ

𝜉= ൬2𝑚𝑐ℏ2 ൰

13 (𝑥− 𝐸𝑐)

𝑑2𝑑𝜉2 𝜓ሺ𝜉ሻ− 𝜉𝜓ሺ𝜉ሻ= 0

§ 4.5.3 Triangular Well

Example:

§ 4.6 Electron confined to atom

See lecture note