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A Wave Packet in a Square Well (An Electron in a Box) Changing with Time. Last Time: Light

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A Wave Packet in a Square Well (An Electron in a Box) Changing with Time. Last Time: Light Electron in a Box A wave packet in a square well (an electron in a box) changing with time. Last Time: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in terms of probability The probability of detecting a photon within a narrow region of width δx at position x is directly proportional to the square of the light wave amplitude function at that point. 2 Prob(in δδx at x) ∝ A(x) x 2 Probability Density Function: Px( ) ∝ A(x) The probability density function is independent of the width, δx , and depends only on x. SI units are m-1. Double Slit: Electrons A light analogy….. There is no electron wave so we assume an analogy to the electric wave and call it the wave function, psi, : Ψ()x The intensity at a point on the screen is proportional to the square of the wave function at that point. 2 Px()= Ψ () x The Probability Density Function is the “Reality”! Probability: Electrons The probability of detecting an electron within a narrow region of width δx at position x is directly proportional to the square of the wave function at that point: 2 Prob(in δδx at x) = Ψ(x) x Probability Density Function: 2 Px( ) = Ψ(x) The probability density function is independent of the width, δx , and depends only on x. SI units are m-1. Note: The above is an equality, not a proportionality as with photons. This is because we are defining psi this way. Also note, P(x) is unique but psi in not since –psi is also a solution. DEGENERACY. If the strip isn’t narrow, then we integrate the probability density function so that the probability that an electron lands somewhere between xL and xR is: xxRR ≤≤ = = ψ 2 Prob(xLR x x ) ∫∫ P() x dx () x dx xxLL dP() x Most Probable: = 0 dx Normalization ∞∞ 2 ∫∫P() x dx= ψ () x dx = 1 −∞ −∞ Electron Waves leads to Quantum Theory 2L Waves: λ =, n =1,2,3..... n n 2 h 1 2 p De Broglie: λ = →=E mv = p 22m hn22 Combine: E = Energy is Quantized! n 8mL2 Wave Packet: Making Particles out of Waves h p = cf= λ λ p= hf/ c Superposition of waves to make a defined wave packet. The more waves used of different frequencies, the more localized. However, the more frequencies used, the less the momentum is known. Heisenberg Uncertainty Principle You make a wave packet by wave superposition and interference. The more waves you use, the more defined your packet and the more defined the position of the particle. However, the more waves you use of different frequencies (energy or momentum) to specify the position, the less you specify the momentum! ∆∆xp > h/4π ∆∆>Et h/4π Little h bar! Which of these particles, A or B, can you locate more precisely? A. A B. B C. Both can be located with same precision. Which of these particles, A or B, can you locate more precisely? A. A B. B C. Both can be located with same precision. Heisenberg Microscope Small wavelength (gamma) of light must be used to find the electron because it is too small. But small wavelength means high energy. That energy is transferred to the electron in an unpredictable way and the motion (momentum) becomes uncertain. If you use long wavelength light ∆x ~λ (infrared), the motion is not as disturbed but the position is uncertain ∆=ph/ λ because the wavelength is too long to see the electron. This results in the ∆∆ > Uncertainty Principle. xp h Electron in a Box The possible wavelengths for an electron in a box of length L. ∆xL~ ∆=p~ p ~/ hλ hL / If you squeeze the walls to decrease ∆x, you increase ∆p! ∆∆xp~/ Lh ⋅ L > h Improved technology will not save us from Quantum Uncertainty! Quantum Uncertainty comes from the particle-wave nature of matter and the mathematics (wave functions) used to describe them! Heisenberg Uncertainty Trying to see what slit an electron goes through destroys the interference pattern. Electrons act like waves going through the slits but arrive at the detector like a particle. Which Hole Did the Electron Go Through? If you make a very dim beam of electrons you can essentially send one electron at a time. If you try to set up a way to detect which hole it goes through you destroy the wave interference pattern. Conclusions: • Trying to detect the electron, destroys the interference pattern. • The electron and apparatus are in a quantum superposition of states. • There is no objective reality. Feynman’s version of the Uncertainty Principle It is impossible to design an apparatus to determine which hole the electron passes through, that will not at the same time disturb the electrons enough to destroy the interference pattern. General Principles Where do the Wave Functions come from??? Solutions to the time-independent Schrödinger equation: 22d ψ − +=Uψ Eψ 2m dx2 OR d 2ψ 2mU( − E) = ψ dx22 Where does that come from??? The Schrödinger Equation Consider an atomic particle with mass m and mechanical energy E in an environment characterized by a potential energy function U(x). The Schrödinger equation for the particle’s wave function is Conditions the wave function must obey are 1. ψ(x) is a continuous function. 2. ψ(x) = 0 if x is in a region where it is physically impossible for the particle to be. 3. ψ(x) → 0 as x → +∞ and x → −∞. 4. ψ(x) is a normalized function. Wavefunction Fun Wave Function: Probability Amplitude: ψ ()x 2 Probability Density: Px()= ψ () x xxRR ≤≤ = = ψ 2 Probability: Prob(xLR x x ) ∫∫ P() x dx () x dx xxLL ∞∞ 2 Normalization ∫∫P() x dx= ψ () x dx = 1 −∞ −∞ ∞∞ 2 Expectation “average” value: 〈〉x=∫∫xP () x dx = xψ () x dx −∞ −∞ dP() x Most Probable: = 0 dx Quantum Cases • Free Particle • Particle in a rigid box • Particle in a finite box “quantum well’ • Quantum Tunneling • Harmonic Oscillator • Hydrogen Atom (Chapter 42) Wave Function of a Free Particle • The wave function of a free particle moving along the x-axis can be written as ψ(x) = Aei(kx-ωt) – A is the constant amplitude – k = 2π/λ is the angular wave number of the wave representing the particle – A free particle must have a sinusoidal wavefunction because it is not confined. • Although the wave function is often associated with the particle, it is more properly determined by the particle and its interaction with its environment – Think of the system wave function instead of the particle wave function Free Particle Problem A free electron has a wave function at t=0 i(5.00 × 1010 x) ψ (x) = Ae where x is in meters. (a) Show that it satisfies the SE. (b) Find its de Broglie wavelength (c) Find its momentum A Particle in a Rigid Box Consider a particle of mass m confined in a rigid, one- dimensional box. The boundaries of the box are at x = 0 and x = L. 1. The particle can move freely between 0 and L at constant speed and thus with constant kinetic energy. 2. No matter how much kinetic energy the particle has, its turning points are at x = 0 and x = L. 3. The regions x < 0 and x > L are forbidden. The particle cannot leave the box. A potential-energy function that describes the particle in this situation is A Particle in a Rigid Box The solutions to the Schrödinger equation for a particle in a rigid box are For a particle in a box, these energies are the only values of E for which there are physically meaningful solutions to the Schrödinger equation. The particle’s energy is quantized. A Particle in a Rigid Box The normalization condition, which we found in Chapter 40, is This condition determines the constants A: The normalized wave function for the particle in quantum state n is A Particle in a Rigid Box Schrödinger Equation Applied to a Particle in a Box • Solving for the allowed energies gives h2 = 2 Enn 2 8mL • The allowed wave functions are given by nπx 2 nπx ψxn ( )= A sin= sin LL L – The second expression is the normalized wave function – These match the original results for the particle in a box HW P.17 Graphical Representations for a Particle in a Box Energy of a Particle in a Box • We chose the potential energy of the particle to be zero inside the box • Therefore, the energy of the particle is just its kinetic energy h2 = 2 = En 2 nn123,,, 8mL • The energy of the particle is quantized The Quantum Jump Quantum Energy States hn22 E = n 8mL2 Energy is Quantized! Only discrete energy states are allowed. Where is the electron between jumps? EXAMPLE :Energy Levels and Quantum jumps QUESTIONS: Finite Potential Wells The wave function in the classically forbidden region of a finite potential well is The wave function oscillates until it reaches the classical turning point at x = L, then it decays exponentially within the classically forbidden region. A similar analysis can be done for x ≤ 0. We can define a parameter η defined as the distance into the classically forbidden region at which the wave function has decreased to e–1 or 0.37 times its value at the edge: Finite Potential Wells The quantum-mechanical solution for a particle in a finite potential well has some important properties: • The particle’s energy is quantized. • There are only a finite number of bound states. There are no stationary states with E > U0 because such a particle would not remain in the well.
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