Question 2.56: Find the first three excited state energies (to five signifi...

Introduction to Quantum Mechanics – Solution Manual [EXP-27105]

Find the first three excited state energies (to five significant digits) for the harmonic oscillator, by wagging the dog (Problem 2.55). For the first (and third) excited state you will need to set $u [0]==0, u ^{\prime}[0]==1$ .)

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The correct values (in Eq. 2.73) are K = 2n + 1 (corresponding to $E_{n}=\left(n+\frac{1}{2}\right) \hbar \omega$ ). I’ll start by \guessing” 2.9, 4.9, and 6.9, and tweaking the number until I’ve got 5 reliable significant digits. The results (see below) are 3.0000, 5.0000, 7.0000. (The actual energies are these numbers multiplied by $\frac{1}{2} \hbar \omega$ .)

$\frac{d^{2} \psi}{d \xi^{2}}=\left(\xi^{2}-K\right) \psi$ (2.73).

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\operatorname{ND Solve}\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-2.9\right) * u[x]==0, u[0]==1\right.\right.$ ,

$\left.u^{\prime}[0]==1\right\}, u[x],\left\{x, 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 0,5\}$ ,

$\text { Plot Range }->\{ -1 ,5\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-2.99999\right) * u[x]==0\right.\right.

$\left.u^{\prime}[0]==1\right\}, u[x],\left\{x, 10^{-8}, 10\right\}$ ,

$\left. u [0]==0, u ^{\prime}[0]=1\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 0,5.5\}$ ,

$\text { Plot Range }->\{ -.1 ,.7\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-4.99999\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==1, u ^{\prime}[0]=0\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 0,6\}$ ,

$\text { Plot Range }->\{ -1.5 ,1.2\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-5.00001\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==1, u ^{\prime}[0]=0\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 0,6\}$ ,

$\text { Plot Range }->\{ -1.5 ,1.2\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-3.00001\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==0, u ^{\prime}[0]==1\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 0,5.5\}$ ,

$\text { Plot Range }->\{ -.5 ,.7\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-4.9\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==1, u ^{\prime}[0]==0\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 0,4\}$ ,

$\text { Plot Range }->\{ -1.5 ,1.2\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-6.9\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==0, u ^{\prime}[0]==1\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 4,5\}$ ,

$\text { Plot Range }->\{ -1 ,.5\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-6.99999\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==0, u ^{\prime}[0]==1\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 6,5\}$ ,

$\text { Plot Range }->\{ -1 ,.5\}]$ ,

$\text { Plot [ Evaluate }[ u [ x ] /$ .

$\text { NDSolve }\left[\left\{u^{\prime \prime} [x]-\left(x^{\wedge} 2-7.00001\right) * u[x]==0\right.\right.$ ,

$\left. u [0]==0, u ^{\prime}[0]==1\right\}, u [ x ],\left\{ x , 10^{-8}, 10\right\}$ ,

$\text { Max Steps }->10000]],\{ x , 6,5\}$ ,

$\text { Plot Range }->\{ -1 ,.5\}]$ ,

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