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"### Exploration of discrete states \n",
"#### J Wang, _UMass Dartmouth_, www.faculty.umassd.edu/j.wang/\n"
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"### 1 Discrete energies (not as weird as it seems) \n",
"- Visually manipulate possible wave function of Schrodiner eqn in a 1D box.\n",
"- Wave function ($A,B,k = $ adjustable parameters): $ \\frac{d^2\\psi}{dx}+ k^2 \\psi=0, \\quad \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k = \\frac{\\sqrt{2mE}}{\\hbar}$\n",
"- Activity: Interactively inspect the wave function to observe that discrete energies are a natural result of finding continuous wave functions. "
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"source": [
"#### Load libraries, initialize, actual templates for my class to use in HW"
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"source": [
"from ipywidgets import interact\n",
"import matplotlib.pyplot as plt\n",
"from numpy import *\n",
"\n",
"a = 1.0 # width\n",
"x = linspace(0, a, 100) # grid"
]
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"source": [
"#### Main visualization $\\quad \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k=n\\pi/a$"
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"def psi(A, B, k):\n",
" wf = A*cos(k*x) + B*sin(k*x)\n",
" return wf\n",
"\n",
"def plotwf(A=1, B=1, n=1.2):\n",
" plt.plot([0, 0, a, a],[1, 0, 0, 1], 'k-') # the well\n",
" wf = psi(A, B, n*pi/a)\n",
" plt.plot(x, wf, 'b-', lw=2) # plot w.f.\n",
" plt.xlabel('x'), plt.ylabel('Wave function')"
]
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"source": [
"Interactive plotting, $ \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k=n\\frac{\\pi}{a}$"
]
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"interactive(children=(FloatSlider(value=1.0, description='A', max=1.0), FloatSlider(value=1.0, description='B'…"
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"source": [
"interact(plotwf, A = (0,1,0.1), B=(0,1,0.1), n=(0,5,.2));"
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"source": [
"### Observations, $ \\psi = A \\cos(kx )+ B \\sin(k x ), \\quad k=n\\pi/a$\n",
"1. \"$A$\" must be 0 for $\\psi(x=0)$ to be zero for arbitrary $B$ and $n$.\n",
"3. \"$B$\" controls scale only (normalization), $\\psi = B \\sin(k x )$.\n",
"3. \"$n$\" must be an integer for $\\psi(x=a)$ to be zero.\n",
"4. Only when $n=$ an integer will $\\psi$ be continuous across the well to the outside where $\\psi=0$.\n",
"5. Discrete $n$, hence discrete energies, are a natural result of obtaining physically acceptable, continuous solutions from an ODE.\n",
"6. Discrete energies are as natural as discrete harmonics on a guitar string from classical mechanics."
]
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