How Does the Virial Theorem Help Determine Position Uncertainty in a C-H Bond?

[jjc43]

Homework Statement

Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H stretching mode. First, look up the transition energy of a C-H stretch (you can pick whichever you like, they are all the same to two significant figures). This will allow you to calculate the energy spacing between the = 0 = 1 states. Second, you can use this information to calculate the ground state energy. Next, use this information and the Virial theorem to calculate the expectation value of the ground state potential energy. Because the potential energy a depends on the square of the position it can be used to calculate the position uncertainty of the bond length. Given that the C-H bond is about 1 Å, comment on the validity of assuming a small vibration amplitude in the Harmonic Oscillator mode

Homework Equations

Transition energy=ħ ω
ω=2πcṽ where ṽ is the wavenumber
Ground state energy= ħ ω/2
Virial theorem: vhat=ax^b 2<T> = b<V>

Notes:

-For the Harmonic Oscilator, <x>= 0
variance: Δx= (<x^2>-<x>^2)^(-1/2)
-Virial Theorem as applied to the Harmonic oscilattor says 2<KE> = 2<V>
Then use <Etotal> = <KE> + <V>

The Attempt at a Solution

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I am not sure about what to use as the wavenumber, I looked at the IR absorption value for the c-h stretch and used 3000cm^-1.

ω=2πcṽ
= 2π(2.998*10^8m/s)*(300000m^-1)
= 5.65*10^14s^-1

Transition energy= ħ ω
= 5.9551*10^-20J

Ground state energy= ħ ω/2
= 2.97755*10^-20J

I am not sure about what to do after this

 

[TSny]

Welcome to PF!

So far, so good. I would suggest that you continue to follow the outline given in the problem statement. What's the next step after finding the ground state energy?