Physics 214 Homework 5: Superposition, Tunneling, and the Uncertainty Principle

5.1: Uncertainty Principle
a. λ_AV = the wavelength given
b. v=x/t  ∴  Δx=cΔt
c. Δp = ℏ/Δx   p_AV = h/λ_AV
   Δp/p_AV = λ_AV/2πΔx
d. Δλ = (Δp/p_AV)*λ_AV

5.2: Superposition
a. Particle in the well described by Ψ_n.
   E = n²E₁
b. Ψ = C1Ψ₁ + C2Ψ₂ + C3Ψ₃
   P(E1)=C1²
   P(E2)=C2²
   P(E3)=C3²
c. <E> = [1²C1² + 2²C2² + 3²C3²]E₁

5.3: Tunneling
a. T=e^-2KL  with  K=sqrt[4π²(V_o-E)/1.505]
b. increase = {[1.505/4&pi²]*[ln(T/10)/-2L]² + E} - V_o
c. increase = {ln(T/10)/(-2K)} - L  with  K=sqrt[4π²(V_o-E)/1.505]

5.4: Superposition Time Dependence
a. E_n = (h²*c²*n²)/(8*mc²*L²)
b. f_Ψn = E_n/h
c. f_Pn = 0
d. f=(E₂-E₁)/h  ∴  t = h/(E₂-E₁)

NOTE ABOUT HOMEWORK B: THESE ANSWERS MAY NOT BE CORRECT. USE THEM AT YOUR OWN RISK.

HomeworkB 05
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An electron is confined in an infinitely deep 1-D box of length L. The first wall is at x=0, the second wall at x=L.

1) The electron is in the n=3 state. At which of these positions is the electron most likely to be found?

 x=L*(5/6)

2) For the electron in the n=5 state, at which of these positions is it least likely to be found?

 x=L*(6/10)

3) Consider the electron in the n=3 state again. What is the probability of finding the electron within +/- d nm of the center of the box. (Note: d nm is a small distance.)

 N²=2/L  and  P=N²Δx  ∴  P=(2/L)*(2d)

4) Suppose the box had only finite depth, but the same L. How would the probability change?

 decrease

5) Consider a wave function of an electron in this infinite square well. The wave function is the following superposition of n = 2 and n = 3 states:

Ψ(x,t) = A (e^iω₂t sink₂x + 2 e^iω₃t sink₃x )
where k_n = h/λ_n, A = the normalization factor, and ω_nh = 2πE_n

The electron energy is measured at time T = h/2(E3-E2). What is the probability that the measurement of the electron energy yields E = E2 ?

0.2