ConceptTests_part3_all

ConceptTests_part3_all - Phys3220 Michael Dubson U.Colorado...

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Unformatted text preview: Phys3220, Michael Dubson U.Colorado at Boulder Quantum I (PHYS 3220) concept questions Operators, Compared to the original ψ (x), the set of numbers {c 1 ,c 2 ,c 3 ,….} contains: A) more information. B) less information. C)the same information D)cannot be determined/depends. 1 A wavefunction ψ (x) has been expressed as a sum of energy eigenfunctions (u n (x)’s): ψ ( x ) = c n u n ( x ) n ∑ If we want the inner product of V with itself, < V | V >, to be positive (for nonzero V ), what should < A | B > be? A) A 1 B 1 +A 2 B 2 B) A* 1 B 1 +A* 2 B 2 C)|A 1 B 1 +A 2 B 2 | D) More than one of these options E) NONE of these makes sense. 1 Consider a complex vector V : Where V 1 and V 2 are complex numbers (they are the “components of V ”) V ⇔ ( V 1 , V 2 ) If f(x) and g(x) are wave functions, and c is a constant, then < c f|g > = ? A) c < f|g > B) c * < f|g > C)|c| < f|g > D)c < f * |g > E) None of these 60 A vector can be written as a column of its components; likewise a vector in Hilbert space (a wave function) can be written as an infinite column of its components in a basis of the u n s: A) B) C) D) E) something else! 59 r A = A x A y A z Ψ = c 1 c 2 c 3 c 4 M The dot product of two vectors A and B is: The inner product of two wavefunctions, r A ⋅ r B = A i B i i = x , y , z ∑ Ψ = c n ψ n n ∑ and Φ = d n ψ n , n ∑ is d x Ψ * Φ = ... ∫ c n d n n ∑ c n d n n ∑ c n 2 d n n ∑ 2 c n 2 + d n 2 ( 29 n ∑ A vector can be written as a column of its components in a basis ; likewise a vector in Hilbert space (a wave function) can be written as an infinite column of its components in a basis of the ψ n s: A) B) C) D) E) zero 59 r A = A x A y A z Ψ = c 1 c 2 c 3 c 4 M The dot product of two vectors A and B is: ( ˆ x , ˆ y , ˆ z ) r A ⋅ r B = A i B i i = x , y , z ∑ The inner product d x Ψ * Φ of wavefunctions ∫ Ψ = d n ψ n n ∑ and Φ = c n ψ n n ∑ is given by d n * c n n ∑ d n c n n ∑ d n 2 c n n ∑ 2 d n 2 + c n 2 ( 29 n ∑ Do you plan to attend today’s Tutorial (on “functions as vectors”) 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. Do the set of all normalized wave functions form a vector space? A) Yes B) No 74 Viewing ψ (x) as a vector in Hilbert space, what role do the c n ’s and u n ’s play?: 1 A wavefunction ψ (x) has been expressed as a sum of energy eigenfunctions (u n (x)’s): ψ ( x ) = c n u n ( x ) n ∑ A) u n ’s are basis vectors, c n’ s are norms of vectors B) u n ’s are components, c n’ s are norms of vectors C) u n ’s are basis vectors, c n’ s are components D) u n ’s are components, c n’ s are basis vectors Viewing | ψ > as a vector in Hilbert space,...
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ConceptTests_part3_all - Phys3220 Michael Dubson U.Colorado...

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