Show that both sides of the Schödinger Equation are equal for all possible x values. What is the value of B?
Show that the wave function ψ(x) = A exp(−x2/2b2) is a solution to the Schödinger Equation for the harmonic oscillator potential, U(x) = (1/2) mω2x2 with energy E = (1/2)ℏω. That is, show that both sides of the Schödinger Equation are equal for all possible x values if b is a particular value. What value of b is it?
Please help...I honestly have no idea where to even start. Thanks!
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-hbar^2 /(2m) d^2 psi(x)/dx^2 + 1/2 m w^2 x^2 psi(x) = 1/2 hbar w psi(x)
is the time-independent Schroedinger equation with U an E given in the problem.
The function psi(x) should obey this equation.
First obtain the second derivative of psi.
The first derivative is -x/b^2 * psi
The second derivative thus is
- 1/b^2 psi + x^2/b^4 psi
Plugging it in gives
[ hbar^2/(2mb^2) - hbar^2 x^2/(2m b^4) + 1/2 m w^2 x^2 - 1/2 hbar w] psi(x) = 0.
Because psi(x) cannot be zero for all x we must have that the x-independent parts inside the brackets must cancel and also the x-dependent terms must cancel for all x.
The first gives
hbar^2/(2mb^2) -1/2 hbar w = 0 => b = sqrt(hbar/(mw))
Also the second gives this value for b.