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ConceptTests_part2_all

# ConceptTests_part2_all - Quantum I(PHYS 3220 concept...

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Quantum I (PHYS 3220) concept questions

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Schrödinger Equation
Consider the eigenvalue equation How many of the following give an eigenfunction and corresponding eigenvalue? I. f(x) = sin(kx), C = +k 2 II. f(x) = exp(-x), C = +1 III. f(x) = exp(i k x), C = -k 2 IV. f(x) = x 3 , C = 6 A) 1 B) 2 C) 3 D) all 4 E) None ) ( )] ( [ 2 2 x f C x f dx d = 31

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Is ? d dt Ψ * ( x , t ) -∞ Ψ ( x , t ) dx = d dt Ψ * ( x , t ) Ψ ( x , t ) ( 29 -∞ dx A)Yes, no problem! B)There’s something not right about this…
Two particles, 1 and 2, are described by plane wave of the form exp[i(kx–ωt)]. Particle 1 has a smaller wavelength than particle 2: λ 1 < λ 2 Which particle has larger momentum? A) particle 1 B) particle 2 C) They have the same momentum D) It is impossible to answer based on the info given. 36

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Ψ 1 (x, t) and Ψ 2 (x, t) are two solutions of the time-dependent SE. Is Ψ sum (x, t) = a Ψ 1 (x, t) + b Ψ 1 (x, t) also a solution of the TDSE? A) Yes, always B) No, never C) Depends on Ψ 1 (x, t) and Ψ 2 (x, t) D) Depends on a and b 32
Ψ 1 (x, t) and Ψ 2 (x, t) are two NORMALIZED solutions of the time- dependent SE. Is Ψ sum (x, t) = a Ψ 1 (x, t) + b Ψ 1 (x, t) also a normalized solution of the TDSE? A) Yes, always B) No, never C) Depends on Ψ 1 (x, t) and Ψ 2 (x, t) D) Depends on a and b 32

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Which expression below would be the QM equation for <KE>? A) B) C) D) None of these! E) More than one! ( 29 x t x t x m k d ) , ( ) , ( 2 * 2 2 - Ψ Ψ - ( 29 x t x x t x m d ) , ( ) , ( 2 2 2 * 2 - Ψ Ψ - ( 29 x t x t x x m d ) , ( ) , ( 2 * 2 2 2 - Ψ Ψ -
After assuming a product form solution Ψ (x,t) = ψ (x)· φ (t), the TDSE becomes If the potential energy function V in the Schrödinger Equation is a function of time, as well as x [V = V(x,t)] would separation of variables still work; that is, would there still be solutions to the SE of the form Ψ (x,t) = ψ (x)· φ (t)? A) Yes, always B) No, never C) Depends on the functional dependence of V on x and t E V x m t i = + - = 2 2 2 1 2 1 ψ ψ ∂ϕ ϕ 30

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Ψ 1 (x, t) and Ψ 2 (x, t) are two solutions of the time-dependent SE. Is Ψ sum (x, t) = a· Ψ 1 (x, t) + b· Ψ 2 (x, t) also a solution of the TDSE? A) Yes B) No C) Depends on Ψ 1 (x, t) and Ψ 2 (x, t) D) Depends on a and b 32
Do you know what the momentum operator is? A) Yes B) No 33

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Do you plan to attend today’s Tutorial (on relating classical to Quantum, and qualitative “sketching” of wave functions) A) Yes, at 3 pm B) Yes, at 4 pm C) Perhaps, more likely at 3 D) Perhaps, more likely at 4 E) No, can’t come/not planning on it.
Given Ψ n (x, t) as one of the eigenstates of Ĥ Ψ n = E n Ψ n , what is the expectation value of the Hamiltonian-squared? A) E n B) E n 2 C) zero D) E n 2 – E n E) Something else/it really depends!! ? d ) H ˆ ( H ˆ ˆ * 2 = Ψ Ψ = x H n n 34

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Ψ 1 and Ψ 2 are two energy eigenstates of the Hamiltonian operator. They are non-degenerate, meaning they have different eigenvalues E 1 and E 2 .
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