Download 1. An electron in an infinite potential well (width A) is in a

1. An electron in an infinite potential well (width A) is in a superposition state given by the
1
wavefunction: 𝜓(𝑥) = [𝜓1 (𝑥) + 𝜓2 (𝑥)]. Here 𝜓1 (𝑥) is the ground state (lowest
√2
energy state) and 𝜓2 (𝑥) is the 1st excited state or next highest energy level of the infinite
potential well.
a. Find the probability density function|Ψ(𝑥, 𝑡)|2 . Remember Ψ(𝑥, 𝑡) = 𝜓(𝑥)𝜑(𝑡)
b. Find the average position (expectation value) of the particle 〈𝑥(𝑡)〉
2. In class we learned about the uncertainty in position and momentum of a particle. Here
we will determine the uncertainty (Δ𝑥Δ𝑝) for a particle confined in an infinite potential
well. V(x) = 0 inside the well (for 0 < x < A) and V(x) = infinity everywhere else. First
determine the expectation value〈𝑥𝑛 〉. Recall that for a general operator T, the expectation
𝐴
value is given by 〈𝑇〉 = ∫0 𝜓𝑛∗ (𝑥)𝑇𝜓𝑛 (𝑥)𝑑𝑥. Next calculate the expectation values 〈𝑥𝑛2 〉,
〈𝑝𝑛 〉, 𝑎𝑛𝑑 〈𝑝𝑛2 〉 . Remember the quantum operator for momentum p is given as 𝑝 =
𝑑
−𝑖ℏ 𝑑𝑥, and the subscripts n indicate the quantized state of the particle (i.e. first energy
eigenstate is E1). Keep your answers in the most general form. Now calculate Δ𝑥𝑛 =
1
1
(〈𝑥𝑛2 〉 − 〈𝑥𝑛 〉2 ) ⁄2 and likewise calculate Δ𝑝𝑛 = (〈𝑝𝑛2 〉 − 〈𝑝𝑛 〉2 ) ⁄2. Finally calculate the
uncertainty (Δ𝑥Δ𝑝). What can you say about the uncertainty of the particle in higher
states (as n increases from 1 to 2 …….)?
−13.6
3. The energy levels for the Hydrogen atom is given as 𝐸𝑛 = 𝑛2 in eV and n is an integer
1,2,3…..etc. Using this expression draw an energy level diagram for the Hydrogen atom.
Derive an expression for the energy (in eV) and wavelength (nm) of light emitted from
transitions between energy levels. What are the three longest wavelengths (in nm) for
transitions terminating at n = 2?
4. Problem 2.25
5. Problem 2.26
6. Problem 2.27
7. Problem 2.29
8. Problem 2.33
9. Problem 2.43