Quantum Superposition and Optical Transitions
Outline
Generating EM Fields Time-Varying Wavefunctions Superposition of Energy States
1 Maxwell and Schrödinger
Maxwells Equations Quantum Field Theory d E · d l = − B · d l … is thought to be the unique and C dt S correct outcome of combining the rules of quantum mechanics with the d H · d l = J · dA + E · dA principles of the theory of relativity. C dt S The Schrodinger Equation The Wave Equation ∂ 2 ∂2 ∂2E ∂2E −j ψ = − ψ y = μ y ∂t 2m ∂x2 2 2 ∂z ∂t (free-particle) Dispersion Relation Dispersion Relation ω2 = c2k2 2k2 ω = ck ω = 2m Energy-Momentum Energy-Momentum p2 E = ω = ck = cp E = 2m (free-particle)
2 P-N Junctions and LEDs
p-type n-type LED
Power Source Resistor Not Shown
High energy electrons (n-type) fall into low energy holes (p-type)
3 P-N Junctions and LEDs
Red Light Yellow Emitted Light Emitted Small Gap Large Gap ENERGY ENERGY
4 P-N Junctions and LEDs
Uncertain energy during transition from high energy to low energy
−iE1t/ −iE2t/ Ψ(x, t)=ψ1(x)e + ψ2(x)e
5 Coupling of Electric and Magnetic Fields
Maxwell’s Equations couple H and E fields..
Oscillating B generates H… Oscillating E generates H… d E · dl = − B · dA H · d l C dt S C d = J · dA + E · dA S dt S
How are the oscillating fields generated ?
6 Time-Dependent Schrodinger Equation
For that matter, how do we get ANYTHING to move ?
states of definite energy
Ψ(x, t)=e−iEt/ψ(x) ∂ 2 ∂2Ψ 2 ∂2ψ i Ψ=− + V (x)Ψ Eψ = − + V (x)ψ ∂t 2m ∂x2 2m ∂x2
Schrodinger says that definite energy states do not move, they are stationary !
|Ψ(x, t)|2 = |ψ(x)|2
7 Example: Superposition of Energy States • It is possible that a particle can be in a superposition of “eigenstates” with different energies.
– Such superpositions are also solutions of the time-dependent SEQ! – What is E of this superposition?
Let’s see how these superpositions evolve with time. • Particle is described by a wavefunction involving a superposition of the two lowest infinite square well states (n=1 and 2)
−iω1t −iω2t Ψ(x, t)=ψ1(x)e + ψ2(x)e ψ(x) E1 E2 ω1 = ω2 = V=∞ V=∞ ψ1 2 E1 = E2 =4E1 8mL2 0 ψ L x π 2π 2 ψ (x)=A sin x ψ2 = A1 sin x 1 1 L L 8 Example: Superposition of Energy States The probability density is given by: |Ψ(x,t)|2 : 2 2 2 |Ψ(x, t)| = |ψ1(x)| + |ψ2(x)| +2ψ1ψ2 cos((ω1 − ω2)t)
Because the cos term oscillates between ±1, |Ψ(x,t)|2 oscillates between: 2 2 2 2 |Ψ(x, t1)| = |ψ1(x)+ψ2(x)| |Ψ(x, t2)| = |ψ1(x) − ψ2(x)| |ψ(x, t )|2 |ψ(x, t )|2 Probability 1 2
0 L x 0 L x
particle localized on left side of well particle localized on right side of well
E − E ω = 2 1 The frequency of oscillation between these two extremes is
9 Numerical Example |ψ(x, t =0)|2 • Consider the numerical example: V=∞ V= An electron in the infinite square well potential ∞ is initially (at t=0) confined to the left side of the well, and is described by the following wavefunction: 0 L x 2 π 2π 2 Ψ(x, t =0)=A sin x + sin x |ψ(x, t0)| L L L V=∞ V=∞
If the well width is L = 0.5 nm, determine the
time to it takes for the particle to “move” to the right side of the well. λn =2L/n 0 L x h2 1.505eV · nm2 E = = E = E n2 n 2 2 n 1 2meλn λn period T = 1/f = 2t0 with f = (E -E )/h 1.505 eV · nm2 1.505 eV · nm2 2 1 E = = =1.505 eV 1 4L2 4(0.5nm)2
−15 T h h 4.136 × 10 eV · sec −16 to = = = = =4.6 × 10 sec 2 2(E2 − E1) 2(3E1) 2(3 × 1.5eV)
10 Example: Superposition of Energy States Consider a particle in an infinite potential well, which at t= 0 is in the state:
−iω2t −iω4t Ψ(x, t)=0.5ψ2(x)e +0.866ψ4(x)e
with ψ2(x) and ψ4(x) both normalized. 2π ψ (x)=A sin x 2 2 L 0 L x 4π ψ (x)=A sin x 4 4 L 1. If we measure the energy of the particle: What is the measured energy?
(a) E2 (b) E4 (c) 0.25 E2 + 0.75 E4 (d) It depends on when we measure the energy
2. If we measure the energy of the particle: What is the expected (average) energy?
(a) E2 (b) E4 (c) 0.25 E2 + 0.75 E4 (d) It depends on when we measure the energy
11 Normalizing Superposition States
• It’s a mathematical fact that any two eigenstates with different eigenvalues (of any measurable, including energy) are ORTHOGONAL » Meaning: ∗ ψ1 (x)ψ2(x)dx =0
So when you normalize a superposition of normalized energy eigenstates, you just have to make the sum of the absolute squares of their coefficients come out 1.
ψ(x) π ψ (x)=A sin x 1 1 L V=∞ V=∞ ψ1 2π ψ (x)=A sin x 2 2 L 0 ψ2 L x
12 Energy of Superposition States
• The important new result concerning superpositions of energy eigenstates is that these superpositions represent quantum particles that are moving. Consider:
−iω1t −iω2t Ψ(x, t)=A1ψ1(x)e + A2ψ2(x)e
• But what happens if we try to measure E on a wavefunction which involves more than one energy? – We can still only measure one of the allowed energies,
i.e., one of the eigenstate energies (e.g., only E1 or E2 in Ψ(x,t) above)!
2 2 If Ψ(x,t) is normalized, |A1| and |A2| give us the probabilities that energies E1 and E2, respectively, will be measured in an experiment!
• When do we not know the energy of an electron ?
13 Beautiful Consistency
• At what frequency does the charge oscillation occur ?
• How much energy does the field take away ?
• What is the energy of the photon that is released ?
Quantum mechanics gives us the oscillating dipole, Maxwell gives us the field !
14 Atomic Transitions
iE1st iE2pt Ψ=c1sφ1se + c2pφ2pe
r
V (r) 2p ΔE 1s
photon
15 Solar Cells and Photodetectors
r V (r) P1
2p ΔE 1s Emission
photon
Classical: Oscillating electric field drives charge oscillation
Quantum: Electric field creates superposition of energy states – which have an oscillating charge density
16 (junction of two differently doped pieces of the same semiconductors) Semiconductor Homojunction Solar Cell
p-type Silicon n-type Silicon
IT IS ENERGETICALLY FAVORABLE FOR electron ELECTRONS TO GO TO THE CONDUCTION MATERIAL ON THE RIGHT BAND
METAL VALENCE IT IS ENERGETICALLY METAL CONTACT BAND FAVORABLE FOR HOLES TO CONTACT STAY IN THE MATERIAL ON hole THE LEFT Resistor A
V
Animated Photogeneration
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6.007 Electromagnetic Energy: From Motors to Lasers Spring 2011
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