- #1

Garlic

Gold Member

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Dear PF community, I am back with a question :)

The solutions for the quantum harmonic oscillator can be found by solving the Schrödinger's equation with:

Hψ = -hbar/2m d²/dx² ψ + ½mω²x² ψ = Eψ

Solving the differential equation with ψ=C exp(-αx²/2)

gives:

-hbar/2m (-α + α²x²)ψ + ½mω²x²ψ = Eψ

(α hbar²/2m - E)ψ + x²(½mω² - hbar²/2m α²)ψ = 0

And the solution in the internet says, that in order the Schrödinger equation to be solveable, the coefficients in the second term (with x²) has to be =0.

And we find the value of α that makes the term =0.

I don't understand: Why does the term with x² have to vanish, in order to make the Schrödinger eq. solveable?And my second question:

we find α=mω/hbar

And with this relation: (α hbar²/2m - E ) =0 we find the Energy E0=½ω hbar.

But how do we find the other energy eigenvalues of the system? For E1=3/2 ω hbar the α term is 3 times larger.

The second term in the Schrödinger equation, (½mω² - hbar²/2m α²) won't be equal to zero if we take any other value for α that isn't =mω/hbar.

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc3.html#c1

Thank you for your time,

garlic

The solutions for the quantum harmonic oscillator can be found by solving the Schrödinger's equation with:

Hψ = -hbar/2m d²/dx² ψ + ½mω²x² ψ = Eψ

Solving the differential equation with ψ=C exp(-αx²/2)

gives:

-hbar/2m (-α + α²x²)ψ + ½mω²x²ψ = Eψ

(α hbar²/2m - E)ψ + x²(½mω² - hbar²/2m α²)ψ = 0

And the solution in the internet says, that in order the Schrödinger equation to be solveable, the coefficients in the second term (with x²) has to be =0.

And we find the value of α that makes the term =0.

I don't understand: Why does the term with x² have to vanish, in order to make the Schrödinger eq. solveable?And my second question:

we find α=mω/hbar

And with this relation: (α hbar²/2m - E ) =0 we find the Energy E0=½ω hbar.

But how do we find the other energy eigenvalues of the system? For E1=3/2 ω hbar the α term is 3 times larger.

The second term in the Schrödinger equation, (½mω² - hbar²/2m α²) won't be equal to zero if we take any other value for α that isn't =mω/hbar.

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc3.html#c1

Thank you for your time,

garlic

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