Wave Function_vibtrating nuclei

[terp.asessed]

Homework Statement

The Ground state wave function governing the motion of a pair of vibrating nuclei looks like:

wave function = wave (x) = (a/pi)^1/4 e^-ax2/2

where a = alpha = mu*w/(h bar), which is determined by mu = reduced mass of the pair
where w = angular vibrational frequency

...So, suppose the vibrating atoms in question were C and O. Compute what v~, the frequency of the vibration in cm-1, would have to be if the root-mean square fluctation in bond length were 0.035Angstom.

I lost track of the units for my solution, and I am not sure if I have done correctly...

Homework Equations

wave function = wave (x) = (a/pi)^1/4 e^-ax2/2

a = alpha = mu*w/(h bar), which is determined by mu = reduced mass of the pair

w = angular vibrational frequency

root mean square = <x²> - <x>²

The Attempt at a Solution

root mean square = <x²> - <x>²

where <x²> = integral (-infinite to infinite) wave(x) x² wave(x) dx
= which I got 1/(2a) as a result

<x> = integral (x=o to L) p(x) x p(x) dx which I got as 0 in the end

so...I input:
root mean square = <x²> - <x>² = 1/(2a) - 0 = 1/(2a)

since
root mean square = 0.035 angstrom = 1/(2a) and b/c a = mu*w/(h bar), I rearranged so that:

w = (h bar)/(0.07mu)...where mu = m₁m₂/(m₁ + m₂)

For mass of 1 C atom: 12.01g/(6.022x10²³ atoms) x 1kg/10³g = 1.99 x 10^-26kg

For mass of 1 O atom: 16.00g/(6.022x10²³ atoms) x 1kg/10³g = 2.66 x 10^-26kg

mu = 1.14 x 10^-20kg

therefore w = (h/(2pi))1/(0.07mu) = 1.32 x 10^-13 omega (is this right?) I know h has units of J*2 and mu = kg...but together they do NOT make omega...

therefore v~ = w/(2pi*c) where c = speed of light = 1.32 x 10^-13 (assuming the unit is correct) / (2pi*3x10⁸m/s) = 7.02 x 10-23 s^-1

b/c I got lost track of units...I am not sure if the units in the solution is right...if someone could (1) explain the units in each calculations I did and then (2) give me hints as how to keep track of units for near future too, that would be so wonderful.

 

[TeethWhitener]

You made a numerical error in your value of $\mu$. If your mass for a carbon atom is ~10^-26kg and your mass for an oxygen atom is ~10^-26kg, there's no way the reduced mass will be ~10^-20kg. Fix that first. As for the units, Planck's constant $\hbar$ is in J s (joule-seconds). Maybe breaking this (and everything else) down into base units (kg, m, s) will help. Just a heads up, you also forgot the "root" part of "root mean square."

 

[terp.asessed]

Hello--thanks you for the reply--so, you're right, I miscalculated: μ = 1.14 x 10^-26 kg

Also, since,
ℏ = J*s = kg*m²/s²...is omega a right unit for w?

w = (h/(2pi))1/(0.07mu) = 1.32 x 10-7 (J*s)/kg = 1.32 x 10-7 m²/s...? Did I make mistake? It is of acceleration unit, not velocity...
Aside, could you please clarify by what you mean by "root part of the root mean square"? If you could, thanks!

 

[TeethWhitener]

You're close with $\hbar$. Joules are kg*m²/s² (you can remember this by recalling that joules are a measure of energy and kinetic energy is $\frac{1}{2}mv^2$), so J*s is kg*m²/s. Also, it's clear that you know that $\mu$ has units of kg, but 0.07 also has units. This comes from the root mean square of distance, which, by the way, is $$\sqrt{\langle x^2 \rangle - \langle x \rangle ^2 }$$ not just $$\langle x^2 \rangle - \langle x \rangle ^2$$ The units of rms distance are the same as the units of plain old distance, which in this case is just angstroms. Hopefully that should get you started.

 

[paranormal]

First of all, great job on attempting to solve this problem! It seems like you have the right approach, but you got a little mixed up with the units.

To clarify, the units for the wave function are not important in this problem. The only important units are for the root mean square fluctuation in bond length, which is given in angstroms.

To keep track of units, it can be helpful to write out the units for each variable in the equation. For example, in this case, the units for the reduced mass (mu) would be kg, the units for the angular vibrational frequency (w) would be s^-1, and so on.

Now, let's go through your calculations step by step to see where the units got mixed up.

1. For the root mean square fluctuation in bond length, you correctly calculated it as 1/(2a) using the equation you provided. However, the units should be in angstroms, not 1/m. So your final answer should be 0.035 angstroms.

2. For the reduced mass (mu), you correctly calculated it as 1.14 x 10^-20 kg. However, when you calculated the mass of the C and O atoms, you used the wrong conversion factor. The correct conversion factor to use would be 1 amu = 1.66 x 10^-27 kg. So the mass of 1 C atom would be 12.01 amu, which is equal to 1.99 x 10^-26 kg. Similarly, the mass of 1 O atom would be 16.00 amu, which is equal to 2.66 x 10^-26 kg. Using these values, you should get a reduced mass of 1.14 x 10^-26 kg, not 1.14 x 10^-20 kg.

3. For the angular vibrational frequency (w), you correctly calculated it as (h bar)/(0.07mu). However, you used the wrong units for h bar. The correct units for h bar are J*s, not J*2. So your final answer for w should be in units of s^-1.

4. Finally, for the frequency (v~), you correctly used the equation v~ = w/(2pi*c). However, you used the wrong units for c. The speed of light (c) is in units of