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Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 5: Q32P (page 245)

Imagine two non interacting particles, each of mass , in the one dimensional harmonic oscillator potential (Equation 2.43). If one is in the ground state, and the other is in the first excited state, calculate $鉄� (x_{1} - x_{2})^{2} 鉄�$ assuming
(a) they are distinguishable particles, (b) they are identical bosons, and (c) they are identical fermions. Ignore spin (if this bothers you, just assume they are both in the same spin state.)

Short Answer

Expert verified

(a) $((x_{1} - x_{2})^{2}) = \frac{2 h}{m 蝇} .$

(b) $(x_{1} - x_{2})^{2} = \frac{h}{m 蝇}$

(c) $(x_{1} - x_{2})^{2} = \frac{3 h}{m 蝇}$

Step by step solution

01

(a) Determining ⟨(x1-x2)2⟩ when they are distinguishable particles.

From the Previous problem :

$鉄� x {鉄�}_{0} = 0 鉄� x {鉄�}_{1} = 0 {(x^{2})}_{0} = \frac{h}{2 m 蝇} x^{2} 1 = \frac{3 h}{2 m 蝇}$

We need to calculate:

$(0 |x| 1) = {鈭�}_{- 鈭�}^{鈭�} x 蠄_{0} (x) 蠄_{1} (x) d x = \sqrt{\frac{h}{2 m 蝇}} (\sqrt{1} 未_{0, 0} + \sqrt{0} 未_{1, - 1}) = \sqrt{\frac{h}{2 m 蝇}}$

From Equation 5.21:

role="math" localid="1658402226570" $({(x_{1} - x_{2})}^{2}) = \frac{h}{2 m 蝇} + \frac{3 h}{2 m 蝇} - 0 = \frac{2 h}{m 蝇}$

02

(b) Determining ⟨(x1-x2)2⟩when they are identical bosons,

From Equation 5.21

$({(x_{1} - x_{2})}^{2}) = \frac{2 魔}{m 蝇} + 2 \frac{h}{2 m 蝇} = \frac{h}{m 蝇}$

03

(c) Determining ⟨(x1-x2)2⟩ when they identical fermions,

From Equation 5.21:

$({(x_{1} - x_{2})}^{2}) = \frac{2 h}{m 蝇} + \frac{2 h}{2 m 蝇} = \frac{3 h}{m 蝇}$

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Most popular questions from this chapter

(a) Construct the completely anti symmetric wave function $蠄 (x_{A}, x_{B}, x_{C})$ for three identical fermions, one in the state $蠄_{5}$ , one in the state $蠄_{7}$ _,and one in the state $蠄_{17}$

(b)Construct the completely symmetric wave function $蠄 (x_{A}, x_{B}, x_{C})$ for three identical bosons (i) if all are in state $蠄_{11}$ (ii) if two are in state $蠄_{19}$ and another one is role="math" localid="1658224351718" $蠄_{1}$ c) one in the state $蠄_{5}$ , one in the state $蠄_{7}$ _,and one in the state $蠄_{17}$

(a) Find the chemical potential and the total energy for distinguishable particles in the three dimensional harmonic oscillator potential (Problem 4.38). Hint: The sums in Equations5.785.78and5.795.79can be evaluated exactly, in this case鈭掆垝no need to use an integral approximation, as we did for the infinite square well. Note that by differentiating the geometric series,

$\frac{1}{1 - x} = {鈭�}_{n = 0}^{鈭�} x^{n}$

You can get

$\frac{d}{d x} (\frac{x}{1 - x}) = {鈭�}_{n = 1}^{鈭�} (n + 1) x^{n}$

and similar results for higher derivatives.

(b)Discuss the limiting caserole="math" localid="1658400905376" $k_{B} T 鈮� h 蝇$ .
(c) Discuss the classical limit,role="math" localid="1658400915894" $k_{B} T 鈮� h 蝇$ , in the light of the equipartition theorem. How many degrees of freedom does a particle in the three dimensional harmonic oscillator possess?

(a) Calculate $< (1 / |r_{1} - r_{2}|) >$ for the state $蠄_{0}$ (Equation 5.30). Hint: Do $d^{3} r_{2}$ integral

first, using spherical coordinates, and setting the polar axis along $r_{1}$ , so

that

$蠄_{0} (r_{1}, r_{2}) = 蠄 100 (r_{1}) 蠄 100 (r_{2}) = \frac{8}{{蟺补}^{3}} e^{- 2 (r_{1} + r_{2}) / a}$ (5.30).

$|r_{1} - r_{2}| = \sqrt{r_{1}^{2} + r_{2}^{2} - 2 r_{1} r_{2} {肠辞蝉胃}_{2}} .$

The $胃_{2}$ integral is easy, but be careful to take the positive root. You鈥檒l have to

break the $r_{2}$ integral into two pieces, one ranging from 0 to $r_{1}$ ,the other from $r_{1} t o 鈭�$

Answer: 5/4a.

(b) Use your result in (a) to estimate the electron interaction energy in the ground state of helium. Express your answer in electron volts, and add it to $E_{0}$ (Equation 5.31) to get a corrected estimate of the ground state energy. Compare the experimental value. (Of course, we鈥檙e still working with an approximate wave function, so don鈥檛 expect perfect agreement.)

$E_{0} = 8 (- 13.6 e V) = - 109 e V$ (5.31).

Use the method of Lagrange multipliers to find the rectangle of largest area, with sides parallel to the axes that can be inscribed in the ellipse ${(\frac{x}{a})}^{2} + {(\frac{y}{b})}^{2} = 1$ . What is the maximum area?

Calculate the Fermi energy for noninteracting electrons in a two-dimensional infinite square well. Let 蟽 be the number of free electrons per unit area.

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