Question

$\textbf{Three-Dimensional Anisotropic Harmonic Oscillator}$. An oscillator has the potential-energy function $U(x, y, z) = {1\over2} k{'_1}(x^2 + y^2) + {1\over 2} k{'_2}z^2$, where $k{'_1} > k{'_2}$. This oscillator is called $anisotropic$ because the force constant is not the same in all three coordinate directions. (a) Find a general expression for the energy levels of the oscillator (see Problem 41.44). (b) From your results in part (a), what are the ground-level and first-excited-level energies of this oscillator? (c) How many states (different sets of quantum numbers $nx, ny,$ and $nz$) are there for the ground level and for the first excited level? Compare to part (c) of Problem 41.44.

Name: Problem 45, Chapter 41: University Physics with Modern Physics
Uploaded: 2023-01-09T15:35:38-08:00
Channel: Lainey Roebuck

   $\textbf{Three-Dimensional Anisotropic Harmonic Oscillator}$. An oscillator has the potential-energy function $U(x, y, z) = {1\over2} k{'_1}(x^2 + y^2) + {1\over 2} k{'_2}z^2$, where $k{'_1} > k{'_2}$. This oscillator is called $anisotropic$ because the force constant is not the same in all three coordinate directions. (a) Find a general expression for the energy levels of the oscillator (see Problem 41.44). (b) From your results in part (a), what are the ground-level and first-excited-level energies of this oscillator? (c) How many states (different sets of quantum numbers $nx, ny,$ and $nz$) are there for the ground level and for the first excited level? Compare to part (c) of Problem 41.44.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Chapter 41, Problem 45

Instant Answer

Solved by Expert Lainey Roebuck

01/09/2023

Step 1

This suggests that the motion in the $x$-$y$ plane and the $z$ direction are independent. The $x$ and $y$ motions are coupled through the same spring constant $k'_1$, while the $z$ motion has a different spring constant $k'_2$. Show more…

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$\textbf{Three-Dimensional Anisotropic Harmonic Oscillator}$. An oscillator has the potential-energy function $U(x, y, z) = {1\over2} k{'_1}(x^2 + y^2) + {1\over 2} k{'_2}z^2$, where $k{'_1} > k{'_2}$. This oscillator is called $anisotropic$ because the force constant is not the same in all three coordinate directions. (a) Find a general expression for the energy levels of the oscillator (see Problem 41.44). (b) From your results in part (a), what are the ground-level and first-excited-level energies of this oscillator? (c) How many states (different sets of quantum numbers $nx, ny,$ and $nz$) are there for the ground level and for the first excited level? Compare to part (c) of Problem 41.44.