1. Concept test 12.1
Suppose at time 𝑡=0, the initial wavefunction of an electron in a 1D infinite square well is Ψ 𝑥 = Ψ 1 𝑥 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 get any of the allowed energies 𝐸 𝑛 , 𝑛=1,2,3…
You can get only 𝐸 1 or 𝐸 2 with equal probability.
You can get only 𝐸= 𝐸 𝐸 2 .
You can get only 𝐸= 1 3 𝐸 𝐸 2 .
None of the above.

2. Concept test 12.2
Suppose at time 𝑡=0, the initial wavefunction of an electron in a 1D infinite square well is Ψ 𝑥 = Ψ 1 𝑥 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 = Ψ 1 𝑥 Ψ 2 𝑥 after a certain time.
The wavefunction Ψ 𝑥 = Ψ 1 𝑥 Ψ 2 (𝑥) will become a delta function immediately after the energy measurement and the system will stay in that state.
The wavefunction will become a delta function immediately after the energy measurement and go back to Ψ 𝑥 = Ψ 1 𝑥 Ψ 2 (𝑥) after a certain time.
None of the above.

3. Concept test 12.3
Suppose at time 𝑡=0, the initial wavefunction of an electron in a 1D infinite square well is Ψ 𝑥 = Ψ 1 𝑥 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 = 𝐸 𝐸 2
𝐸 = 1 3 𝐸 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

4. Concept test 12.4
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 = Ψ 1 𝑥 Ψ 2 𝑥 .
Ψ 𝑥 = Ψ 1 𝑥 − Ψ 2 𝑥
Ψ 𝑥 = Ψ 1 𝑥 + 𝑖 Ψ 2 𝑥
Ψ 𝑥 = 𝑖 Ψ 1 𝑥 + 𝑖 Ψ 2 𝑥
A. 1 only B. 2 only C. 3 only D. 1 and 3 only E. none of the above

5. Concept test 12.5
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)
A. 1 only B. 2 only B. 3 only C. 1 and 2 only E. All of the above

6. Concept test 12.6
Choose all of the following statements that are correct about an electron in an eigenstate of the position operator.
The position of the electron is well defined in this state.
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.
A. 1 only B. 1 and 2 only C. 1 and 3 only D. 2 and 3 only E. All of the above.