A Three-Dimensional Isotropic Harmonic Oscillator. An...

$\textbf{A Three-Dimensional Isotropic Harmonic Oscillator}$. An isotropic harmonic oscillator has the potentialenergy function $U(x, y, z) = {1\over2}k'(x^2 + y^2 + z^2)$. ($Isotropic$ means that the force constant $k'$ is the same in all three coordinate directions.) (a) Show that for this potential, a solution to Eq. (41.5) is given by $\psi = \psi_{n_x}(x)\psi _{n_y}(y)\psi _{n_z}(z)$. In this expression, $\psi _{n_x}(x)$ is a solution to the one-dimensional harmonic-oscillator Schrodinger equation, Eq. (40.44), with energy $E_{n_x} = (n_x + {1\over2} )\hslash \omega$. The functions $\psi _{n_y}(y)$ and $\psi_{n_z}(z)$ are analogous one-dimensional wave functions for oscillations in the $y$- and $z$-directions. Find the energy associated with this $\psi$. (b) From your results in part (a) what are the ground-level and first-excited-level energies of the three-dimensional isotropic oscillator? (c) Show that there is only one state (one set of quantum numbers $n_x, n_y,$ and $n_z$) for the ground level but three states for the first excited level.

Transcript

00:02 Okay, so in this problem we have a potential energy described by u that is equal one half of k x square plus y square plus z square and in the first item we have to show that the wave equation psi equals psi x of x of x of x that multiplies psi -y of y, psi -z of z, is in fact a solution for the time -independent shrugting the equation.

00:53 Okay? so let's remember here what is the time -independent shrugly in this equation.

00:59 So it's just negative -blank constant square to m that multiplies.

01:07 D, d, x, plus d, d, y, plus d, d, z, apply to the wave function psi, plus upsi, that is equal the energy psi.

01:35 So we just need to show that this is in fact a solution.

01:41 And then describe the energies, okay? so how we're going to do this? well, it's pretty simple.

01:50 Let's simply apply the psi, which is this guy in here, and substitute and each one of these components here, of these terms in here.

02:05 So after doing this we will get a expression that is huge, so let's just write the expression here.

02:15 So, so, so we have negative h to m, d, psi, x, dx, dx, plus one half k x square, psi x that multiplies z and psi y.

02:44 And we must repeat this for all the three components.

02:51 So this is going to be plus negative.

02:56 2m d psi y divided by d y plus one half of k y square psi y x and finally the last one h2 m d psi z plus 1d plus 1st z half of k z square, si z, psi x, x, psi y.

03:45 Okay, so this huge equation here is the shorting the equation, just the left side, okay? i'm not considering the sides, the size is the side of the energy, so the left side.

04:00 And as we can see, we have in here three terms, three equations inside a parentheses, okay, therefore we can attribute that each one of them is going to be equal some defined energy, which is ez, psi z, ey, psi, y, and finally e x, psi -x.

04:34 Okay, therefore this is the entire schroenter equation.

04:40 So here is how we prove that this function psi, here, generic function, is in fact a solution of the time independent showing this equation.

04:52 Okay, because we can simply describe this solution as being negative h2 to m...

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