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C 1. Which of the following phenomena could be explained by classical physics and did not require a quantum hypothesis in order to make theory agree with experiment? (a) (b) (c) (d) Blackbody spectra (e) Diffraction of light (f) Low-temperature heat capacity in (g) perfect crystals The photoelectric effect (h) 2. If a normalized wave packet Ψ is given as ! ( x, y, z,t ) = Gravity (b) and (e) (a), (b), and (d) None of the above # $ cn " n ( x, y, z)e %iE t / h , n n =1 what is the probability that an experiment will cause the system to collapse to the specific stationary state j? (a) |cj|2 (b) (c) (d) Quantum mechanics does not allow (e) you to know this probability One (f) < ψj | H | ψj > (g) cj* (h) 3. Which of the following statements about the de Broglie wavelength λ are true? (a) λ=h/p (b) λ decreases as mass increases if (f) velocity is constant λ increases as momentum increases (g) λ is a constant, like Planck's (h) constant (c) (d) (e) cj (b) and (d) Only Schrödinger's cat knows A particle that has zero velocity has an infinite de Broglie wavelength All of the above (a), (b) and (e) (c) and (d) 4. Which of the below equations can be false for an arbitrary pair of orthonormal functions f and g? (a) (b) (c) (d) < |f|2 >< |g|2 > = 1 <f|H|g>=0 <f|g>=0 fg = 0 (e) (f) (g) (h) f*g – g*f = 0 (a) and (c) (b), (d) and (e) All of the above NAME: ________________________________________________________________ 2 5. Which of the below expectation values are or may be non-zero? (a) (b) (c) < sinx | x | cosx > < sin2x | x | cos2x > < f | [A,B] | g > where A and B commute < Ψm | H | Ψn > where Ψm and Ψn are non-degenerate stationary states (d) (e) (f) (g) <f | g > – < g | f >* <µ mn> for a forbidden transition (a), (d) and (e) (h) (b), (c), and (f) 6. Which of the following did Bohr assume in order to derive a model consistent with the photoemission spectra of one-electron atoms? (a) The electron is a delocalized wave (b) (d) The ionization potential is equal to (f) the work function The one-electron atom is like a (g) particle in a box The Coulomb potential is quantized (h) 7. Which of the following statements about a well behaved wave function is true? (a) It must be continuous (e) (b) It may take on complex values (f) (c) (d) It must be quadratically integrable (g) It must be equal to its complex (h) conjugate (c) (e) The angular momentum of the electron is quantized (a) and (b) (b) and (d) None of the above Its square modulus has units of probability density It must be an eigenfunction of the momentum operator (d) and (f) (a), (b), (c), and (e) NAME: ________________________________________________________________ 3 8. Which of the following statements are false about the free particle? (a) Its Schrödinger equation " h2 d 2 % $! ! E ' ( (x ) = 0 2 # 2m dx & (b) It may be regarded as having a (f) wave function that is the superposition of a left-moving particle and a right-moving particle Its energy levels are all non- (g) negative Its energy levels are quantized (h) (c) (d) is (e) Valid wave functions include where ! ( x ) = Aeikx + Be"ikx 2mE k= h Valid wave functions include ! ( x ) = N coskx where N is a normalization constant and k is defined in (e) above All of the above None of the above 9. Given a particle of mass m in a box of length L having the wave function 2 # n"x % , what is the energy of the level corresponding to n = 4? !( x ) = sin$ L L & (a) Since this wave function is not an eigenfunction of the Hamiltonian the question cannot be answered < Ψ | px2 | Ψ > 16 times the energy of the ground state 8" 2 h 2 mL2 (b) (c) (d) ! 10. (e) 8h2 / mL2 (f) (g) (c) and (d) (b) and (e) (h) None of the above On which of the below functions does the parity operator Π act in the fashion Π[f(x)] = (–1)f(x)? (a) x (e) (b) (c) (d) x3 eix sinx (f) (g) (h) Any eigenfunction Hamiltonian (b) and (d) (a), (b), and (d) (b), (d), and (e) of the NAME: ________________________________________________________________ 4 Short-answer (20 points) Prove that, given a pair of normalized but not orthogonal functions ψ1 and ψ2, the function ψ3 = ψ2 – Sψ1 is orthogonal to ψ1 if S is the overlap integral of ψ1 and ψ2. Is ψ3 normalized? (Use the back of the page if necessary). We evaluate "1 "3 = "1 "2 # S"1 = "1 "2 # S "1 "1 = S # S •1 =0 which proves the orthogonality (we use normalization of ψ1 in going from line 2 to 3). ! With respect to normalization, we evaluate "3 "3 = "2 # S"1 "2 # S"1 = "2 "2 # 2S "1 "2 + S 2 "1 "1 = 1# 2S • S + S 2 = 1# S 2 so ψ3 is only normalized for the boring case of S = 0 (which means that ψ1 and ψ2 were orthogonal to begin with since no change to ψ2 is required to generate ψ3). ! NAME: ________________________________________________________________