Physics 401: Quantum Mechanics I Chapter 4

Physics 401: Quantum Mechanics I Chapter 4

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3-D Schroedinger Equation

The ground state energy of the particle in a 3D box is ( 1 2 +1 2 +1 2 ) π2 2 2ma 2 = 3ε. What is the energy of the 1st excited state? A) 4ε B) 5ε C) 6ε D) 8ε E) 9ε 83

In the 3D infinite square well, what is the degeneracy of the energy corresponding to the state (n x, n y, n z ) = (1, 2, 3)? A) 1 B) 3 C) 4 D) 6 E) 9 84

Spherical coordinates

In Cartesian coordinates, the volume element is dx dy dz. In spherical coordinates, the volume element is A) r 2 sinθ cosφ dr dθ dφ B) sinθ cosφ dr dθ dφ C) r 2 cosθ sinφ dr dθ dφ D) r sinθ cosφ dr dθ dφ E) r 2 sinθ dr dθ dφ 99

A planet is in elliptical orbit about the sun. P A The torque, about the sun is: τ = r F on the planet 95 A) Zero always B) Non-zero always C) Zero at some points, non-zero at others.

P A The magnitude of the angular momentum of the planet about the sun L = r p A) Greatest at the perihelion point, P B) Greatest at the aphelion point, A C) Constant everywhere in the orbit is: 96

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In Cartesian coordinates, the normalization condition is In spherical coordinates, the normalization dx integral has limits of integration: dy dz Ψ 2 =1. + 0 A) dr 2π 0 π dθ dϕ B) dr 0 + 2π 0 dθ π 0 dϕ + 0 C) dr 2π 0 2π dθ dϕ D) dr 0 + π 0 dθ π 0 dϕ E) None of these 100

A and B are positive constants. r is radial distance (0 r < ). Using the whiteboards: Sketch A r and B r 2 What does the graph y(r) = B r 2 A r look like? 111

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Hydrogen Energy Levels

Recall, for hydrogen: E n = E 1 n 2, with E 1 = (ke2 ) 2 m e /2 2 Consider He+ (1 e- around a nucleus, Q= 2e). If you look at Balmer lines (e- falling from higher n down to n=2) what part of the spectrum do you expect the emitted radiation to fall in? A) Visible B) IR C) UV D) It s complicated, not obvious at all.

The spectrum of "Perkonium" has only 3 emission lines. 5 ev 3 ev 2 ev E(eV) 200 300 400 500 600 wavelength(nm) Which energy level structure is consistent with the spectrum? (A) (B) (C) (D) 2 3 5 2 4 5 5 5 7 7 7 8 114 10

As indicated in the figure, the n = 2, l = 0 state and the n = 2, l = 1 state happen to have the same energy (given by E 2 = E 1 /2 2 ). Do these states have the same radial wavefunction R(r)? V eff l = 1 l = 2 x 4 n = 3 n = 2 l = 0 n = 1 A) Yes B) No

Radial probability: ground state of hydrogen

Consider an electron in the ground state of an H- atom. The wavefunction is ψ (r) = Aexp( r / a ) 0 Where is the electron more likely to be found? A) Within dr of the origin (r = 0) B) Within dr of a distance r = a 0 from the origin? r = a 0 y A B x 120

A. Yes B. No Are you here today? C. Eigenfunction, eigenvalue, eigenfunction, eigenvalue what s the difference again?

Hydrogen atom orbitals

We are solving the equation 2 2m d 2 u " dr 2 + $ ke2 # r + 2 l(l +1) 2mr 2 % 'u = Eu & What, then, is the full 3-D wave function for hydrogen atom stationary states? A) u(r,θ,φ) B) u(r)y lm (θ,φ) C) ru(r)y lm (θ,φ) D) r 2 u(r)y lm (θ,φ) E) None of these

True (A) or False (B) Any arbitrary stationary state of an electron bound in the H-atom potential can always be written as ψ n,l,m (r,θ,φ) = R nl (r)y l m (θ,φ), with suitable choice of n, l, and m.

Suppose at t=0, Ψ(r,t = 0) = 1 2 (ψ 210 +ψ 200 ) Is Ψ (r,t) given very simply by Ψ (r,t =0) e -iet/ħ? A)Yes, that s the simple result B) No, it s more complicated (a superposition of two states with different t dependence => sloshing )

Suppose at t=0, Ψ(r,t = 0) = 1 2 (ψ 200 +ψ 300 ) Is Ψ (r,t) given very simply by Ψ (r,t =0) e -iet/ħ? A)Yes, that s the simple result B) No, it s more complicated (a superposition of two states with different t dependence => sloshing )

Allowed 1-photon transitions

How many of the following transitions to the 2p in an H-atom will result in emission of a photon? E s p d f 4 3 2 n = 1 A) all of them: 11 B) None of them: 0 C) 8 D) 9 E) 6

Midterm results P402 W16 Exam Distribution 13 12 11 10 Average = 70 9 Frequency 8 7 6 5 4 3 2 1 0 0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100+ Score Midterm Final

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The angular momentum operator is L ˆ = r ˆ p ˆ = ( L ˆ, L ˆ, L ˆ ) x y z with e.g. L ˆ = y ˆ p ˆ z ˆ p ˆ,..., x z y L ˆ = x ˆ p ˆ y ˆ ˆ z y p x Is L ˆ Hermitian? (Hint: Is L z Hermitian?) A) Yes B) No C) Only L z is (L x and L y are not) D) L z is not (but L x and L y are) E) Are you joking here? Can I do this as a clicker question?

Is the commutator, [ x ˆ, p ˆ ] y zero or non-zero? A) Zero B) Non-zero 86

The commutator, [ y ˆ p ˆ, x ˆ ˆ ] z p z zero or non-zero? A) Zero B) Non-zero C) Sometimes zero, sometimes non-zero 97

The commutator, [ L 2, L ˆ ] z z zero or non-zero? A) Zero B) Non-zero C) Sometimes zero, sometimes non-zero 98

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Angular momentum raising and lowering operators

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Postulates of Quantum Mechanics (1) The state of a particle is completely represented by a normalized vector in Hilbert space, which we call ψ> (2) All physical observables, Q, are associated with Hermitian operators Q, and the expectation value of Q in some state ψ> is <ψ Q ψ> (3) A measurement of Q on a particle in state ψ> is certain to return a particular value, λ, iff ("if and only if") Q ψ> = λ ψ>, (i.e. if and only if ψ> is already an eigenvector of Q, with eigenvalue λ) (3a) If you measure Q in any state ψ>, you are certain to obtain one of the eigenvalues of Q. The probability of measuring some eigenvalue λ is given by <u λ ψ> 2, where u λ > is defined to be the eigenvector of Q with eigenvalue λ (3b) After a measurement gives you the value λ, the system will collapse into the state u λ > (4) The time evolution of the state ψ> is given by Schrodinger's equation: i ψ t = Ĥ ψ

= 2

A. Yes B. No Are you here today? C. It depends on what kind of spin you put on it.

Stern-Gerlach Experiment (W. Gerlach & O. Stern, Z. Physik 9, 349-252 (1922).

Schematic of Stern-Gerlach experiment

Stern-Gerlach experiments #1 and 2 State preparer State analyzer #1 #2

Stern-Gerlach experiment #3

Stern-Gerlach experiment #4