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Q4: We now know that the quantized energy levels of a harmonic oscillator (H-O) are $E_v \propto \hbar \omega (v + \frac{1}{2})$ ($v = 0, 1, 2, \dots$) with the first eigenfunction/eigenstate at $E_{v=0}$ $\psi_0 = N_0 e^{-\frac{m\omega}{2\hbar}x^2}$ Please show that the normalization factor in the above is $N_0 = \left( \frac{m\omega}{\pi \hbar} \right)^{\frac{1}{4}}$ Use simple hand drawing to show the energy levels of H-O in its potential V(x) and the eigenfunction on each energy level up to the first 5 levels. Only pay attention to the number of lobes and nodes for each state. Put the number of nodes for each eigenfunction on the side. Also note the exponential term (Gaussian function) governed decay of the eigenfunction into the classical forbidden area with $E_v(x) < V(x)$, which is called "tunneling" in quantum mechanics.

Name: q4 we now know that the quantized energy levels of a harmonic oscillator h o are ev propto hbar omega v frac12 v 0 1 2 dots with the first eigenfunctioneigenstate at ev0 psi0 n0 e fracmo 45899
Uploaded: 2024-07-30T00:37:37-08:00
Duration: 4 min 16 s
Channel: Adi S
Description: q4 we now know that the quantized energy levels of a harmonic oscillator h o are ev propto hbar omega v frac12 v 0 1 2 dots with the first eigenfunctioneigenstate at ev0 psi0 n0 e fracmo 45899

          Q4: We now know that the quantized energy levels of a harmonic oscillator (H-O) are
$E_v \propto \hbar \omega (v + \frac{1}{2})$ ($v = 0, 1, 2, \dots$) with the first eigenfunction/eigenstate at $E_{v=0}$
$\psi_0 = N_0 e^{-\frac{m\omega}{2\hbar}x^2}$
Please show that the normalization factor in the above is
$N_0 = \left( \frac{m\omega}{\pi \hbar} \right)^{\frac{1}{4}}$
Use simple hand drawing to show the energy levels of H-O in its potential V(x) and the eigenfunction on each energy level up to the first 5 levels. Only pay attention to the number of lobes and nodes for each state. Put the number of nodes for each eigenfunction on the side. Also note the exponential term (Gaussian function) governed decay of the eigenfunction into the classical forbidden area with $E_v(x) < V(x)$, which is called "tunneling" in quantum mechanics.

$q4 we now know that the quantized energy levels of a harmonic oscillator h o are ev propto hbar omega v frac12 v 0 1 2 dots with the first eigenfunctioneigenstate at ev0 psi0 n0 e fracmo 45899$

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