Chapter 40: Q. 23 (page 1175)

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

Short Answer

Expert verified

$\frac{d^{2} Ψ_{1} (x)}{d x^{2}} = - \frac{2 m}{ℏ^{2}} (E - \frac{1}{2} k x^{2}) Ψ_{1} (x)$

Step by step solution

Step 1. Given information

The Schrödinger wave equation for a quantum harmonic oscillator is,

$\frac{d^{2} Ψ}{d x^{2}} = - \frac{2 m}{ℏ^{2}} (E - \frac{1}{2} k x^{2}) Ψ (x)$

Here,

E= energy of the harmonic oscillator,

k= force constant

Step 2. for wave function

consider,

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

First derivative,

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

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

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

The second derivative,

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

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

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

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

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

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

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

$= - (\frac{1}{{(\sqrt{\frac{ℏ}{m ω}})}^{2}} - \frac{x^{2}}{{(\sqrt{\frac{ℏ}{m ω}})}^{4}}) Ψ_{1} (x)$

$= - (\frac{m ω}{ℏ} - \frac{m^{2} ω^{2} x^{2}}{ℏ^{2}}) Ψ_{1} (x)$

$= - (\frac{m ω}{ℏ} - \frac{m^{2} (k / m) x^{2}}{ℏ^{2}}) Ψ_{1} (x) (As, ω^{2} = k / m)$

$= - \frac{2 m}{ℏ^{2}} (\frac{1}{2} ℏ ω - \frac{1}{2} k x^{2}) Ψ_{1} (x)$

The ground state energy of the harmonic oscillator is $\frac{1}{2} ℏ ω$ .

Therefore,

$\frac{d^{2} Ψ_{1} (x)}{d x^{2}} = - \frac{2 m}{ℏ^{2}} (E - \frac{1}{2} k x^{2}) Ψ_{1} (x)$

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

Short Answer

Step by step solution

Step 1. Given information

Step 2. for wave function

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

One App. One Place for Learning.

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Verify that the n=1 wave function ψ1(x) of the quantum harmonic oscillator really is a solution of the Schrödinger equation. That is, show that the right and left sides of the Schrödinger equation are equal if you use the ψ1(x) wave function.

Short Answer

Step by step solution

Step 1. Given information

Step 2. for wave function

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Atoms and Radioactivity

Classical Mechanics

Solid State Physics

Geometrical and Physical Optics

Oscillations

Study anywhere. Anytime. Across all devices.

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