Figure 6.11 shows a plot of the 10th excited state probability density, |ψ10|2. Mathematica has the Hermite polynomials built-in. The quantum oscillator wave functions are given in equation 6.57; these wave functions are not normalized. The α in these equations is 𝑚 𝜔ℏ(HW Problem 6.36 and in-class work). The argument of the Hermite polynomials in equation 6.57 is listed as “x” but you will want to use 𝑢 = √𝛼𝑥 as the argument when you are actually write down or program the Hermite polynomials.
(a) Write down the (un-normalized) wave function for the 10th excited state; you can write it in terms of α. Also write down the energy for this state (write this energy in terms ℏ and ω)?
This type of which energy act on the energy eigenstates of the harmonic oscillator potential producing a un-normalized state of higher or lower energy.
a± =1/√2m(~/i∂/∂x ± imωx)
A=- ℏ^2 d^2/
(b) Plot ψ10 and |ψ10|2(use u rather than x for your independent variable); your |ψ10|2 plot should look like Figure 6.11.
(c) Normalize ψ10 (use u);
Normalization the stationary wave functions are r a 1 2 2 ψn (x) = 2n√π n! Hn (ax) e− a x 2 .The diodes are available in the normalized E24 ±1 % (BZX84-A), ±2 % (BZX84-B) and approximately ±5 % (BZX84-C) tolerance range. The series includes 37 breakdown voltages with nominal working voltages from 2.4Vto75 V.
(d) Find the probability that the electron is in the region −0.5 ≤ √𝛼𝑥 ≤ 0.5. Use 3 significant figures for these numerical answers.
(e) What is 〈𝛼𝑥2〉 for this excited state?