Harmonic oscillator's energy levels

AI Thread Summary
The discussion centers on determining the energy levels and probabilities of a particle in a one-dimensional harmonic oscillator described by a specific wave function. The energy levels are derived from the formula E_n = (n + 1/2)ħω, and the wave function is expressed as a superposition of eigenstates, suggesting the presence of only two relevant energy levels due to the cubic term in the wave function. Participants discuss the process of finding coefficients for the eigenstates using orthogonality and integration, while also addressing the normalization constant. The conversation highlights the complexity of connecting the given wave function to the eigenstates and calculating the probabilities associated with each energy level. Ultimately, the participants are working towards understanding the relationship between the wave function and the harmonic oscillator's energy states.
  • #51
Good! You still have a few typos, and it would have been easier to rewrite the lefthand side in terms of ##\xi## to get
$$\sqrt{\frac{8}{15}} \sqrt[4]{\frac{m\omega}{\hbar\pi}} \, \xi^3 e^{-\xi^2/2} = \sqrt[4]{\frac{m\omega}{\hbar\pi}} \, e^{-\xi^2/2} \left[a_0 + \frac{1}{\sqrt{2}} a_1 (2\xi) + \frac{1}{\sqrt{8}} a_2 (4\xi^2-2) + \frac{1}{\sqrt{48}} a_3 (8\xi^3-12\xi)\right]$$ or, equivalently,
$$\sqrt{\frac{8}{15}} \, \xi^3 = a_0 + \frac{1}{\sqrt{2}} a_1 (2\xi) + \frac{1}{\sqrt{8}} a_2 (4\xi^2-2) + \frac{1}{\sqrt{48}} a_3 (8\xi^3-12\xi).$$ Can you take it from there?
 
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  • #52
Yes, thank You:D I finally calculated the coefficients ##a_1=\sqrt{\frac{3}{5}}## and ##a_3=\sqrt{\frac{2}{5}}##. I hope that his time they are correct. Now I need to calculate the energies corresponding to those eigen functions by solving schrodinger equation, and coefficients squared are their probabilities, right?
 
  • #53
Those look reasonable. The squares of the coefficients are indeed the probabilities; however, while it's a good exercise to go through at least once, you should not have to solve the Schrodinger equation for this problem. It's probably already done in your textbook. You should be familiar with the basic results.
 
  • #54
of course You were right, I forgot about the most basic formula for energy levels:P Now there is last part of the problem-calculate the probability of finding the particle in the classically forbidden region- I thought about using this formula: ##P_n=2 \int_|x_n|^{+\infty} |\psi(\xi)|^2 \,d\xi. ##
 
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  • #55
about the second part of the problem: I tried to calculate the probability of finding a particle of energy ##e_3=\frac{3}{2}\hbar\omega## using a formula ##P(E=E_1)=\frac{2}{\sqrt{\pi}2^1 1!}\int_\sqrt{3}^{+\infty} exp(-\xi^2){H_1}^2(\xi)\,dx##, but the result comes out as 0,06696. Where am I making a mistake?
 
  • #56
Could someone please tell me if the formula I chose is right?
 
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