Harmonic Oscillator Negative Energy(Quantum)

[Relativeee]

Homework Statement

Ok so the question is, is the state u(x) = Bxe^[(x^2)/2] an energy eigenstate of the system with V(x) = 1/2*K*X^2 and what is the probability per unit length of this state.

Homework Equations

The Attempt at a Solution

So the way i did this was, to find if the state is an energy eigenstate of the system i checked to see if it satisfied the time independent Schrödinger equation. After all of the Calculations , came up with, 3- 2E/hw = 0, and the definition of E is E = (n+1/2)hw, so the only way for this solution to be satisfied is if the energy is negative, -2 to be exact. So with this I am pretty sure that this cannot be an energy eigenstate of the system since the harmonic oscillator has to have a positive energy(unless there is a specific exception i do not know about) and therefore we cannot find the probability density because the wave-function is not normalizable(blows up at inifinity) and that's what is bugging me, is it as simple as that or am i missing something, like for instance is there a way to normalize this wave-function so as to find the probability density?

 

[diazona]

The Attempt at a Solution

So the way i did this was, to find if the state is an energy eigenstate of the system i checked to see if it satisfied the time independent Schrödinger equation. After all of the Calculations , came up with, 3- 2E/hw = 0, and the definition of E is E = (n+1/2)hw, so the only way for this solution to be satisfied is if the energy is negative, -2 to be exact.

No it isn't. Try that again.

 

[kuruman]

After all of the Calculations , came up with, 3- 2E/hw = 0, and the definition of E is E = (n+1/2)hw, so the only way for this solution to be satisfied is if the energy is negative, -2 to be exact.

That's not very exact. If 3 - 2 E/hbar*ω = 0 and you solve for E, what do you get? Hint: Move -2 E/hbar*ω over to the right side and change its sign.

 

[Relativeee]

Oh my goodness i apologize, i meant to put 3 + 2E/hw

 

[azimos]

So, given your definition of E, n must equal to -2. However, n has to be 0 or positive integer.
Does that mean this is the energy eigenstate of the system?

 

[diazona]

Oh my goodness i apologize, i meant to put 3 + 2E/hw

OK, in that case, we can perhaps help you if you show the work that led you to that expression.