Quantum Mechanics IV Quiz “The most incomprehensible thing about the world is that it is comprehensible.”– Albert Einstein Quiz

Q24.0 The name of this differential equation is: Einstein’s last equation deBroglie’s broken equation Space-independent Schroedinger Equation Time-independent Schroedinger Equation Bound-state Schroedinger’s Equation

Q24.0 The name of this differential equation is: Einstein’s last equation deBroglie’s broken equation Space-independent Schroedinger Equation Time-independent Schroedinger Equation Bound-state Schroedinger’s Equation Describes a free particle of mass m

Q24.1 A solution to this differential equation is: A cos(kx) A e-kx A sin (kx) (B & C) (A & C)

Q24.1 A solution to this differential equation is: A cos(kx) A e-kx A sin (kx) (B & C) (A & C) Ans: E – Both (A) and (C) are solutions. solution if

Q24.2 …makes sense, because Condition on k is just saying that (p2)/2m = E. V=0, so E= KE = ½ mv2 = p2/2m The total energy of the electron is: Quantized according to En = (constant) x n2, n= 1,2, 3,… Quantized according to En = const. x (n) Quantized according to En = const. x (1/n2) Quantized according to some other condition but don’t know what it is. Not quantized, energy can take on any value.

Q24.2 …makes sense, because Condition on k is just saying that (p2)/2m = E. V=0, so E= KE = ½ mv2 = p2/2m The total energy of the electron is: Quantized according to En = (constant) x n2, n= 1,2, 3,… Quantized according to En = const. x (n) Quantized according to En = const. x (1/n2) Quantized according to some other condition but don’t know what it is. Not quantized, energy can take on any value. Ans: E - No boundary, energy can take on any value.

Q24.3 An electron with definite momentum in free is given by: if k positive x = 0 x = L Compare the probability of electron being in dx at x = L is _______ probability of being in dx at x = 0. Probability at x=L > probability at x=0 for all time The two probabilities are always the same C. Probability at x=L < probability at x=0 for all time D. oscillates up and down in time between bigger and smaller E. Without being given k, can’t figure out

Q24.3 An electron with definite momentum in free is given by: if k positive x = 0 x = L Compare the probability of electron being in dx at x = L is _______ probability of being in dx at x = 0. Probability at x=L > probability at x=0 for all time The two probabilities are always the same C. Probability at x=L < probability at x=0 for all time D. oscillates up and down in time between bigger and smaller E. Without being given k, can’t figure out Ans: B - Prob ~ ψ*ψ = A2cos2(kx-ωt) +A2sin2(kx-ωt)= A2, so constant and equal, all x, t.

Which free electron has more kinetic energy? A) 1 B) 2 C) Same big k = big KE 1. small k = small KE 2. if V=0, then E= Kinetic Energy. So first term in Schöd. Eq. is always just kinetic energy! Curvature  KE

Infinite Potential Well (“particle in a box”) x<0, V(x) ~ infinite x> L, V(x) ~ infinite 0<x<L, V(x) =0 Clever approach means just have to solve: Energy with boundary conditions, ψ(0)=ψ(L) =0 x L Solution a lot like microwave & guitar string

 k=nπ/L Infinite Potential Well (“particle in a box”) ∞ 0 eV 0 L functional form of solution: Apply boundary conditions x=0  ?  A=0 x=L  kL=nπ (n=1,2,3,4 …)  k=nπ/L 1 2 What is the momentum, p?

Infinite Potential Well (“particle in a box”) E quantized by B. C.’s What is E? A. can be any value (not quantized). B. D. C. E. Screwy rep…. Energy vs. space..! Does this L dependence make sense?

What is potential energy of electron in the lowest energy state (n=1)? Results: Quantized: k=nπ/L Quantized: L Real(ψ) n=1 L What is potential energy of electron in the lowest energy state (n=1)? a. E1 b. 0 c. ∞ d. could be anything For n=4 Note two different energies plotted KE and PE! KE Correct answer is b! V=0 between 0 and L (We set it!)  So electron has KE = E1.

Infinite Potential Well (“particle in a box”) Quantized: k=np/L Quantized: energies are “eigenvalues” wave functions “eigenfunctions” Screwy rep…. Energy vs. space..! Exercise: Sketch wavefunction for the n=5 state and find the energy

Sketch wavefunction for the n=5 state and find the energy Worked out on white board with class. Skipped next several slides to normalization of wavefucntion (Total probability of finding the particle anywhere = 1) Preview: Qualitatively compare the eigen-energy spectrum for “particle in a box”, harmonic oscillator, and Coulomb potential. Lecture ends.

Infinite Potential Well (“particle in a box”) Quantized: k=np/L Quantized: y n=2 What you get quantum mechanically: Electron can only have specific energies. (quantized) What you expect classically: Electron can have any energy Electron is localized Screwy rep…. Energy vs. space..! Electron is delocalized … spread out between 0 and L

Infinite Potential Well (“particle in a box”) Probability of finding particle at a specific x-location? Probability of finding particle at a specific x-location? Screwy rep…. Energy vs. space..!

Infinite Potential Well (“particle in a box”) Probability of finding particle at a specific x-location? Probability of finding particle at a specific x-location? Screwy rep…. Energy vs. space..!

Infinite Potential Well (“particle in a box”) Screwy rep…. Energy vs. space..!

Infinite Potential Well (“particle in a box”) Probability of finding particle at a specific x-location? What is probability in these 3 cases of finding particle at X = L/2 ? Screwy rep…. Energy vs. space..!

Infinite Potential Well (“particle in a box”) Probability of finding particle at a specific x-location? What is probability in these 3 cases of finding particle at X = L/2 ? What is the height in “y” ?? Screwy rep…. Energy vs. space..! How do we do this?

Infinite Potential Well (“particle in a box”) Probability of finding particle at a specific x-location? Screwy rep…. Energy vs. space..!

Infinite Potential Well (“particle in a box”) Probability of finding particle at a specific x-location? Screwy rep…. Energy vs. space..!

Infinite Potential Well (“particle in a box”) iClicker Check: What is probability in these 3 cases of finding particle at X = L/2 ? N = 1 greatest N = 3 greatest Same N = 2, for sure Screwy rep…. Energy vs. space..!

Finite Potential Well (“particle in a box”) What happens if the walls are no longer infinite? Screwy rep…. Energy vs. space..!

Finite Potential Well (“particle in a box”) What happens if the walls are no longer infinite? Screwy rep…. Energy vs. space..! Particle can “leak out” (‘tunnel’) Wavelength is longer than infinite potential case

Finite Potential Well (“particle in a box”) What happens if the walls are no longer infinite? Screwy rep…. Energy vs. space..! Energy levels?  Energy levels are lower than the infinite potential case

For next time More Quantum Mechanics Read material in advance Concepts require wrestling with material