Alpha in the schrodinger's equation for the harmonic oscillator

In summary, the conversation discusses the use of the variable alpha in an equation and how it can be simplified through substitutions. The speaker questions the use of alpha and suggests an alternative substitution. They also mention that there may be multiple ways to solve the equation.
  • #1
asdf1
734
0
why for that equation does alpha= 2E/(hf)?
 
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  • #2
Since you haven't mentioned what alpha is supposed to represent, how are we to supposed to answer that question? I can't read minds you know. Alpha is not an agreed upon universal variable.
To me it sounds like a definition. There to simplify notation.
It is equal to the ration of the energy to the ground state energy.
 
Last edited:
  • #3
sorry for being unclear~
psi`` + (2m/H^2)(E-0.5kx^2)psi=0
my textbook says that to simply that equation, it makes the subsitution
alpha= 2E/(hf) so that
psi``+(alpha-y^2)=0
but i think that it should make the subsitution y= (2m/H^2)E
 
  • #4
asdf1 said:
but i think that it should make the subsitution y= (2m/H^2)E

So go ahead and try it and see if it helps better than the other substitution! It's certainly possible that it might work. There's usually more than one way to skin a cat (or solve an equation), after all. :smile:
 
  • #5
you're right~ :)
just curious why the book made that substituition, though~
 

1. What is the significance of the alpha term in the Schrodinger's equation for the harmonic oscillator?

The alpha term in the Schrodinger's equation for the harmonic oscillator represents the strength of the restoring force that brings the oscillator back to its equilibrium position. It is a measure of the stiffness or rigidity of the oscillator's potential energy function.

2. How does changing the value of alpha affect the behavior of the harmonic oscillator?

Changing the value of alpha in the Schrodinger's equation for the harmonic oscillator will change the frequency and amplitude of the oscillator's motion. A larger alpha value will result in a stiffer oscillator with a higher frequency and smaller amplitude, while a smaller alpha value will result in a more flexible oscillator with a lower frequency and larger amplitude.

3. Is the alpha term constant or does it change over time?

The alpha term in the Schrodinger's equation for the harmonic oscillator is typically considered to be constant, as it represents the inherent properties of the oscillator. However, there are cases where the alpha term may change over time, such as when the oscillator is subject to external forces or when the potential energy function is not purely harmonic.

4. How is the alpha term related to the energy levels of the harmonic oscillator?

The alpha term is directly related to the energy levels of the harmonic oscillator. It determines the spacing between adjacent energy levels, with a larger alpha value resulting in a larger energy spacing between levels. This can be seen from the solution to the Schrodinger's equation, which shows that the energy levels are proportional to (n+1/2) multiplied by the square root of the alpha term.

5. Can the alpha term be negative in the Schrodinger's equation for the harmonic oscillator?

No, the alpha term in the Schrodinger's equation for the harmonic oscillator must be positive. This is because a negative alpha value would result in an upside-down potential energy function, which is not physically realistic. Furthermore, the solution to the Schrodinger's equation would result in imaginary energy levels, which have no physical interpretation.

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