Concept test 14.1 Is the function graph d below a possible wavefunction for an electron in a 1-D infinite square well between 𝑥=0 and 𝑥=𝑎 at time 𝑡=0? It is a possible wavefunction. It is not a possible wavefunction because it does not satisfy the Time-independent Schroedinger Equation. It is not possible because it is neither symmetric nor anti-symmetric about the center of the well. It is not a possible wave function because it does not satisfy the boundary conditions of the system. None of the above

Concept Test 14.2 Choose all of the following statements that are correct about the wave function shown below for an electron interacting with an infinite square well of width 𝑎 between 𝑥=0 and 𝑥=𝑎 at time 𝑡=0. Ψ(𝑥) and 𝜕Ψ 𝜕𝑥 are continuous everywhere. It is a possible wave function because it is a continuous, smooth and normalizable function that satisfies the boundary conditions. It is a possible wave function, and can be obtained by a superposition of energy eigenfunctions according to Fourier series analysis. It is not a possible wave function because it is neither symmetric nor antisymmetric about the center of the well. (I) only B. (II) only C. (III) only D. (I) and (II) only E. None of the above.  

Concept Test 14.3 An electron is interacting with a one dimensional finite square well with a wave function Ψ 𝑥,0 at t = 0. Choose all of the following statements that are correct: Ψ(𝑥,0) 2 must be symmetric about the center of the well. Ψ(𝑥,0) must reflect the symmetry of the potential energy well. Any single-valued, smooth normalizable function is a possible wavefunction Ψ(𝑥,0). (I) only B. (II) only C. (III) only D. (I) and (III) only E. (II) and (III) only

Concept Test 14.4 Choose all of the following statements that are correct about the wave function shown below for an electron interacting with a finite square well of width 𝑎 (V 𝑥 =− 𝑉 0 when 0<𝑥<𝑎 and V 𝑥 =0 anywhere else) at time 𝑡=0: It is a possible wave function because it is the first excited state. It is a possible wave function because it is anti-symmetric about the center of the well. It is not a possible wave function because it goes to zero at the boundaries of the well. It is not a possible wave function because its derivative is not continuous at the boundaries of the well. (I) only B. (III) only C. (IV) only D. (I) and (II) only E. (III) and (IV) only

Smooth function that goes to zero within the region x=0 and x=a. Concept Test 14.5 Choose all of the following statements that are correct about the wave function shown below for an electron interacting with a finite square well of width a (V 𝑥 =− 𝑉 0 when 0<𝑥<𝑎 and V 𝑥 =0 anywhere else). Ψ(𝑥) and 𝜕Ψ 𝜕𝑥 are continuous and single valued everywhere. Smooth function that goes to zero within the region x=0 and x=a. It is a possible wave function because it is a continuous, smooth and normalizable function that satisfies the boundary conditions. It is not a possible wave function because it doesn’t satisfy the boundary conditions; it goes to zero inside the well. It is not a possible wave function because the probability of finding the particle outside the finite square well is zero but quantum mechanically it must be nonzero. A. (I) only B. (II) only C. (III) only D. (II) and (III) only E. None of the above.

Concept test 14.6 Choose all of the following wave functions that are possible wave functions for an electron in a one dimensional infinite square well of width 𝑎 (boundaries between 𝑥=0 and 𝑥=𝑎) at time 𝑡=0: (I) only (I) and (II) only (I) and (III) only (II) and (III) only All of the above

V(x) x I II III Concept Test 14.7 Choose all of the following statements are true about a particle interacting with a double delta function potential energy barrier, as shown below: It is possible to find the particle in all three regions. If the particle starts out in region II, then the particle is in a bound state. The energy levels for this system are discrete. A. (I) only B. (II) only C. (III) only D. (II) and (III) only E. None of the above V(x) x 𝛿(𝑥+𝑎) 𝛿(𝑥−𝑎) 𝑉 𝑥 = 𝑉 0 𝛿(𝑥+𝑎) 𝑉 𝑥 = 𝑉 0 𝛿(𝑥−𝑎) I II III

Concept test 14.8 Suppose at time 𝑡=0, the initial wavefunction of an electron in a 1D infinite square well is Ψ 𝑥 = 1 3 Ψ 1 𝑥 + 2 3 Ψ 2 (𝑥), where Ψ 1 𝑥 and Ψ 2 (𝑥) are the ground state and first excited state wavefunctions. Choose all of the following statements that are correct if you measure the energy at 𝑡=0. You can obtain any of the allowed energies 𝐸 𝑛 , 𝑛=1,2,3… You can obtain only 𝐸 1 or 𝐸 2 with equal probability. You can obtain only 𝐸= 1 3 𝐸 1 + 2 3 𝐸 2 . You can obtain only 𝐸= 1 3 𝐸 1 + 2 3 𝐸 2 . None of the above.

Concept test 14.9 Suppose at time 𝑡=0, the initial wavefunction of an electron in a 1D infinite square well is Ψ 𝑥 = 1 3 Ψ 1 𝑥 + 2 3 Ψ 2 (𝑥), where Ψ 1 𝑥 and Ψ 2 (𝑥) are the ground state and first excited state wavefunctions. If you measure the energy at time t=0, which one of the following statements is correct? The wavefunction will become either Ψ 1 𝑥 or Ψ 2 (𝑥) immediately after the measurement and the system will remain in that state at future times. The wavefunction will become either Ψ 1 𝑥 or Ψ 2 (𝑥) immediately after the measurement and the system will go back to Ψ 𝑥 = 1 3 Ψ 1 𝑥 + 2 3 Ψ 2 𝑥 after a long time. The wavefunction will become either Ψ 1 𝑥 or Ψ 2 (𝑥) immediately after the measurement and the system will evolve into a state which is superposition of all the stationary states after a long time. The wavefunction Ψ 𝑥 = 1 3 Ψ 1 𝑥 + 2 3 Ψ 2 (𝑥) will become a delta function immediately after the energy measurement and the system will stay in that state for all future times. The wavefunction will become a delta function immediately after the energy measurement and go back to Ψ 𝑥 = 1 3 Ψ 1 𝑥 + 2 3 Ψ 2 (𝑥) after a certain time.

Concept test 14.10 Suppose at time 𝑡=0, the initial wavefunction of an electron in a 1D infinite square well is Ψ 𝑥 = 1 3 Ψ 1 𝑥 + 2 3 Ψ 2 (𝑥), where Ψ 1 𝑥 and Ψ 2 (𝑥) are the ground state and first excited state wavefunctions. Choose all of the following statements that are correct about the expectation value of the energy of the system at t=0. 𝐸 = 1 3 𝐸 1 + 2 3 𝐸 2 𝐸 = 1 3 𝐸 1 + 2 3 𝐸 2 𝐸 = −∞ ∞ Ψ ∗ 𝑥 𝐻 Ψ 𝑥 𝑑𝑥 A. 1 only B. 2 only C. 3 only D. 1 and 3 only E. 2 and 3 only

Concept test 14.11 Suppose Ψ 1 𝑥 and Ψ 2 (𝑥) are the ground state and first excited state wavefunctions of an electron in a 1D infinite square well of width a. Choose all of the following functions that represent the same state as Ψ 𝑥 = 1 2 Ψ 1 𝑥 + 1 2 Ψ 2 𝑥 . Ψ 𝑥 = 1 2 Ψ 1 𝑥 − 1 2 Ψ 2 𝑥 Ψ 𝑥 = 1 2 Ψ 1 𝑥 + 𝑖 2 Ψ 2 𝑥 Ψ 𝑥 = 𝑖 2 Ψ 1 𝑥 + 𝑖 2 Ψ 2 𝑥 A. 1 only B. 2 only C. 3 only D. 1 and 3 only E. none of the above

Concept test 14.12 Suppose Ψ(𝑥) is an eigenfunction of the position operator 𝑥 with eigenvalue 𝑥 ′ . Choose all of the following statements that are correct. 𝑥 Ψ 𝑥 = 𝑥 ′ Ψ(𝑥) Ψ 𝑥 =𝛿 𝑥− 𝑥 ′ 𝐻 Ψ 𝑥 =𝐸Ψ(𝑥) 𝐻 Ψ 𝑥 =𝑥′Ψ(𝑥) 1 only B. 3 only C. 4 only D. 1 and 2 only E. 1, 2, and 4 only

Concept test 14.13 Choose all of the following statements that are correct about an electron in an eigenstate of the position operator. The energy of the electron is well defined because he position operator commutes with the Hamiltonian. The momentum of the electron cannot be well defined in this state due to the uncertainty principle. The measurement of position will yield a definite value with 100% certainty. 1 only B. 1 and 2 only C. 1 and 3 only D. 2 and 3 only E. All of the above.