Calculating Force Constant for a Harmonic Oscillator: Units and Conversions

[Lamebert]

Homework Statement

A molecule is a harmonic oscillator, and has a fundamental frequency of 1000 cm⁻¹ and a reduced mass of 10.0 amu. From this, determine the magnitude of the force constant, and express it in units of aJ ˚A⁻² (atto joules per angstrom).

Homework Equations

ω=√(k/m_r)

The Attempt at a Solution

My question is about the requested units. Force constants of harmonic oscillators are given as force*distance, no? So the request for the units in energy/distance² doesn't make sense. Further, using the relevant equation given I end up with amu/cm², or mass/distance². I suppose wavenumbers can be counted as energy as well. Even then, you'd end up with J^2*amu or J*amu/cm. The problem, then, is the remaining mass. I'm fairly certain I'm misunderstanding something in this, however. Any help would be appreciated.

Edit: Apologies for having to move my thread. This is this first part of a multistep quantum mechanical problem. I guess this part is technically more introductory.

 

[vela]

No. The force constant is $k$ as in $F=-kx$ in Hooke's law.

 

[Lamebert]

No. The force constant is $k$ as in $F=-kx$ in Hooke's law.

Sorry, I meant force/distance. My question still stands.

 

[vela]

Energy has units of force times distance, so…

 

[Lamebert]

Energy has units of force times distance, so…

I know but the problem is the units of this problem itself. I need the units to be in energy/distance^2 which is proper but the output of the equation includes mass and energy (amu and wavenumbers^2) which, from what I can tell, can't be converted to any form of a force constant, which is why I'm sure I'm doing something wrong.

Pulling apart the given equation, it tells us that k = ω²m_r. Omega is in wave numbers which is an energy unit, and m_r is a reduced mass in units of amu. So E²* mass (from the equation) should be equal to energy/distance² (from the requested unit in the problem). This is clearly not true, unless Energy is equal to inverse distance² times mass.

 

[vela]

Sorry, I didn't notice the non-standard units for the frequency. That seems like a typo, where it should say the units are s^-1, or you're supposed to follow some sort of convention like mapping wave numbers to frequency via a relation like $c = \omega k$.

 

[vela]

From reading http://quantum.bu.edu/notes/QuantumMechanics/HarmonicOscillator.pdf, I gather 1000 cm^-1 refers to the reciprocal wavelength $\tilde{\nu} = 1/\lambda = \nu/c$, where $\nu$ is the frequency in Hz and $c$ is the speed of light.

 

[Lamebert]

From reading http://quantum.bu.edu/notes/QuantumMechanics/HarmonicOscillator.pdf, I gather 1000 cm^-1 refers to the reciprocal wavelength $\tilde{\nu} = 1/\lambda = \nu/c$, where $\nu$ is the frequency in Hz and $c$ is the speed of light.

So, in other words, your suggestion is multiplying wavenumbers by the speed of light, which will result in a normal frequency (s^-1). Even if this is done, I end up with mass/s² which is... force/distance I guess, which is correct.

Ok, cool. Thanks.

 

[Lamebert]

From reading http://quantum.bu.edu/notes/QuantumMechanics/HarmonicOscillator.pdf, I gather 1000 cm^-1 refers to the reciprocal wavelength $\tilde{\nu} = 1/\lambda = \nu/c$, where $\nu$ is the frequency in Hz and $c$ is the speed of light.

Hmm, one more question. The final answer I get is in units of kg/s². It's true that, when the distances are canceled in the final unit requested in the problem, that these units are correct. How do I apply these distances to the number? For instance, let's say I got 1 kg/s². Can I just multiply it by 1 meter² for sake of unit conversion to get to joule, and divide 10^-20 for the angstrom part?