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Chapter 40: Q. 23 (page 1175)

Verify that the n=1 wave function $蠄_{1} (x)$ of the quantum harmonic oscillator really is a solution of the Schr枚dinger equation. That is, show that the right and left sides of the Schr枚dinger equation are equal if you use the $蠄_{1} (x)$ wave function.

Short Answer

Expert verified

$\frac{d^{2} 唯_{1} (x)}{d x^{2}} = - \frac{2 m}{{鈩�}^{2}} (E - \frac{1}{2} k x^{2}) 唯_{1} (x)$

Step by step solution

01

Step 1. Given information

The Schr枚dinger wave equation for a quantum harmonic oscillator is,

$\frac{d^{2} 唯}{d x^{2}} = - \frac{2 m}{{鈩�}^{2}} (E - \frac{1}{2} k x^{2}) 唯 (x)$

Here,

E= energy of the harmonic oscillator,

k= force constant

02

Step 2. for wave function

consider,

$唯_{1} (x) = A_{1} e^{- \frac{x^{2}}{2 b^{2}}}$

First derivative,

$\frac{d 唯_{1} (x)}{d x} = \frac{d}{d x} (A_{1} e^{- \frac{x^{2}}{2 b^{2}}})$

$= A_{1} \frac{d}{d x} (e^{- \frac{x^{2}}{2 b^{2}}})$

$= - \frac{A_{1}}{b^{2}} x e^{- \frac{x^{2}}{2 b^{2}}}$

The second derivative,

$\frac{d^{2} 唯_{1} (x)}{d x^{2}} = \frac{d}{d x} (\frac{d 唯_{1} (x)}{d x})$

$= - \frac{A_{1}}{b^{2}} \frac{d}{d x} (x e^{\frac{x^{2}}{2 b^{2}}})$

$= - \frac{A_{1}}{b^{2}} e^{- \frac{x^{2}}{2 b^{2}}} + \frac{A_{1}}{b^{4}} x^{2} e^{- \frac{x^{2}}{2 b^{2}}}$

$= - (\frac{1}{b^{2}} - \frac{x^{2}}{b^{4}}) A_{1} e^{- \frac{x^{2}}{2 b^{2}}}$

$= - (\frac{1}{b^{2}} - \frac{x^{2}}{b^{4}}) 唯_{1} (x) (As, 唯_{1} (x) = A_{1} e^{- \frac{x^{2}}{2 b^{2}}})$

03

Step 3 Substituting ℏmω = b in d2Ψ1(x)dx2=-1b2-x2b4Ψ1(x).

$\frac{d^{2} 唯_{1} (x)}{d x^{2}} = - (\frac{1}{b^{2}} - \frac{x^{2}}{b^{4}}) 唯_{1} (x)$

$= - (\frac{1}{{(\sqrt{\frac{鈩�}{m 蝇}})}^{2}} - \frac{x^{2}}{{(\sqrt{\frac{鈩�}{m 蝇}})}^{4}}) 唯_{1} (x)$

$= - (\frac{m 蝇}{鈩�} - \frac{m^{2} 蝇^{2} x^{2}}{{鈩�}^{2}}) 唯_{1} (x)$

$= - (\frac{m 蝇}{鈩�} - \frac{m^{2} (k / m) x^{2}}{{鈩�}^{2}}) 唯_{1} (x) (As, 蝇^{2} = k / m)$

$= - \frac{2 m}{{鈩�}^{2}} (\frac{1}{2} 鈩� 蝇 - \frac{1}{2} k x^{2}) 唯_{1} (x)$

The ground state energy of the harmonic oscillator is $\frac{1}{2} 鈩� 蝇$ .

Therefore,

$\frac{d^{2} 唯_{1} (x)}{d x^{2}} = - \frac{2 m}{{鈩�}^{2}} (E - \frac{1}{2} k x^{2}) 唯_{1} (x)$

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Most popular questions from this chapter

Tennis balls traveling faster than $100 m p h$ routinely bounce off tennis rackets. At some sufficiently high speed, however, the ball will break through the strings and keep going. The racket is a potential-energy barrier whose height is the energy of the slowest string-breaking ball. Suppose that a $100 g m$ tennis ball traveling at $200 m p h$ is just sufficient to break the $2.0 n m$ -thick strings. Estimate the probability that a $120 m p h$ ball will tunnel through the strings without breaking them. Give your answer as a power of $10$ rather than a power of $e$ .

For the quantum-well laser of Figure 40.16, estimate the probability that an electron will be found within one of the GaAlAs layers rather than in the GA As layer. Explain your reasoning.

Use the data from Figure 40.24 to calculate the first three vibrational energy levels of a $C = O$ carbon-oxygen double bond.

A particle of mass m has the wave function $蠄 (x) = A x e x p (\frac{- x^{2}}{a^{2}})$ when it is in an allowed energy level with $E = 0$ .

a. Draw a graph of $蠄 (x)$ versus $x$ .

b. At what value or values of $x$ is the particle most likely to be found?

c. Find and graph the potential-energy function $U (x)$ .

Show that the normalization constant $A_{n}$ for the wave functions of a particle in a rigid box has the value given in Equation 40.26.

See all solutions

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