Quantum Mechanics III Quiz Find solution(s) to the 1-D Schrodinger Equation for a free particle

Q23.1 A 2

Q23.1 A 3

Q23.2 B (note C is wrong !) 4

Q23.2 B (note C is wrong !) 5

Q23.3 C 6

Worked out on White Board Q23.3 Worked out on White Board C 7

Review: Matter Waves de Broglie proposed that matter can be described with waves. Experiments demonstrate the wave nature of particles, where the wavelength matches de Broglie’s proposal. 8

Matter Waves de Broglie wavelength: Find the wavelength of an electron beam of 25 eV: 9

Plane Waves Plane waves (sines, cosines, complex exponentials) extend forever in space: Different k’s correspond to different energies, since 15

Superposition 16

Plane Waves vs. Wave Packets Which one looks more like a particle? 17

How to generalize the Schrodinger Equation to describe non-free particle?

Most physical situations, like H atom, no time dependence in V! Simplification #1:V = V(x) only. (works in 1D or 3D) (Important, will use in all Shrödinger equation problems!!) Ψ(x,t) separates into position part dependent part ψ(x) and time dependent part ϕ(t) =exp(-iEt/ħ). Ψ(x,t)= ψ(x)ϕ(t) Plug in, get equation for ψ(x) “time independent Schrodinger equation”

Lecture ended here. No time to go over remaining slides. Next lecture: Examine effect of boundary conditions on eigen-energies - Example “Particle in a box”

Matter Waves Describe a particle with a wave function. Wave does not describe the path of the particle. Wave function contains information about the probability to find a particle at x, y, z & t. In general: x L -L Simplified: & 22

Matter Waves x L -L Wavefunction L -L x Probability Density Probability of finding particle in the interval dx is Normalized wave functions 23

Matter Waves Assume that a neutron is described by the following wave function for. At what value of x is the neutron most likely to be found? A) XA B) XB C) XC D) There is no one most likely place 24

An electron is described by the following wave function:  L a b c d dx x How do the probabilities of finding the electron near (within dx) of a,b,c, and d compare? d > c > b > a a = b = c = d d > b > a > c a > d > b > c 25

An electron is described by the following wave function:  L a b dx x c d How do the probabilities of finding the electron near (within dx) of a,b,c, and d compare? d > c > b > a a = b = c = d d > b > a > c a > d > b > c 26

Superposition If and are solutions to a wave equation, then so is Superposition (linear combination) of two waves We can construct a “wave packet” by combining many plane waves of different energies (different k’s). 27

Plane Waves vs. Wave Packets For which type of wave are the position (x) and momentum (p) most well-defined? x most well-defined for plane wave, p most well-defined for wave packet. p most well-defined for plane wave, x most well-defined for wave packet. p most well-defined for plane wave, x equally well-defined for both. x most well-defined for wave packet, p equally well-defined for both. p and x are equally well-defined for both. 28

Plane Waves vs. Wave Packets For which type of wave are the position (x) and momentum (p) most well-defined? x most well-defined for plane wave, p most well-defined for wave packet. p most well-defined for plane wave, x most well-defined for wave packet. p most well-defined for plane wave, x equally well-defined for both. x most well-defined for wave packet, p equally well-defined for both. p and x are equally well-defined for both. 29

Uncertainty Principle Δx small Δp – only one wavelength Δx medium Δp – wave packet made of several waves Δx large Δp – wave packet made of lots of waves 30

Uncertainty Principle In math: In words: The position and momentum of a particle cannot both be determined with complete precision. The more precisely one is determined, the less precisely the other is determined. What do (uncertainty in position) and (uncertainty in momentum) mean? 31

Uncertainty Principle A Statistical Interpretation: Measurements are performed on an ensemble of similarly prepared systems. Distributions of position and momentum values are obtained. Uncertainties in position and momentum are defined in terms of the standard deviation. 32

Uncertainty Principle A Wave Interpretation: Wave packets are constructed from a series of plane waves. The more spatially localized the wave packet, the less uncertainty in position. With less uncertainty in position comes a greater uncertainty in momentum. 33

what is V(r,t) for electron interacting with proton? + - -ke2/r, where r is the distance from electron to origin. -ke2/r where r is distance between + and - . Impossible to tell unless we know how electron is moving, because that determines the time dependence. (-ke2/r) x sin(ωt) Something else… Ans: B - Although potential energy will be different as electron moves to different distance, at any given distance will be same for all time. So V(r,t) = V(r) = -ke2/r. H atom.

Solving the Schrödinger equation for electron wave in 1-D: 1. Figure out what V(x) is, for situation given. 2. Guess or look up functional form of solution. 3. Plug in to check if ψ’s and all x’s drop out, leaving an equation involving only a bunch of constants. 4. Figure out what boundary conditions must be to make sense physically. 5. Figure out values of constants to meet boundary conditions and normalization: 6. Multiply by time dependence ϕ(t) =exp(-iEt/ħ) to have full solution if needed. STILL TIME DEPENDENCE! |ψ(x)|2dx =1 -∞ ∞

Where does the electron want to be? The place where V(x) is lowest. V(x) Electrons always tend toward the position of lowest potential energy, just like a ball rolling downhill. x

L Solving Schrod. equ. Before tackling well, understand simplest case. Electron in free space, no electric fields or gravity around. 1. Where does it want to be? 2. What is V(x)? 3. What are boundary conditions on ψ(x)? No preference- all x the same. Constant. None, could be anywhere. Smart choice of constant, V(x) = 0!

What does this equation describe? A. Nothing physical, just a math exercise. B. Only an electron in free space along the x-axis with no electric fields around. C. An electron flying along the x-axis between two metal plates with a voltage between them as in photoelectric effect. D. An electron in an enormously long wire not hooked to any voltages. E. More than one of the above. Ans: E – Both (B) and (D) are correct. No electric field or voltage means potential energy constant in space and time, V=0.

For next time Quantum Mechanics Homework #7 available Midterm #2 Read material in advance Concepts require wrestling with material Homework #7 available Due Monday, Mar. 13th Midterm #2 Monday, Mar. 20th Reviews toward end of next week