Ground state wavelength and uncertainty Bohr/deBroglie model

In summary, Louis de Broglie attempted to explain Bohr's hydrogen atom electron orbits as being circles of just the right circumference for constructive interference to occur, providing an explanation for angular momentum quantization. In the ground state of hydrogen, the wavelength of this wave is 0.332nm. According to the uncertainty principle, the uncertainty in wavelength is 2 times the radius of the atom. However, the model may not make sense as the wavelength decreases as the electron moves farther from the nucleus, which goes against the expected decrease in electrostatic force.
  • #1
Summer95
36
0
A check my work question...

Homework Statement


Louis de Broglie tried to explain Bohr’s hydrogen atom electron orbits as being circles of just the
right circumference such that an electron of the Bohr energy going around the circle will
interfere constructively with itself. This seems to give a nice explanation of the angular momentum quantization. The attempt at a solution

Q1: what would be the wavelength of such a wave in the ground state of hydrogen?

## \lambda n=2\pi r## from the Bohr energy equation we can tell n=1 gives the lowest energy state. We also have an equation for the possible values of r which reduces to ##r=a_{0}n^{2}## and so in our case ##\r=a_{0}##. Therefore ## \lambda n=2\pi r## becomes ## \lambda =2\pi a_{0} =0.332nm ##. Where ##a_{0}=0.0529##.

Q2: According to the uncertainty principle, what would be the uncertainty in the wavelength, given that the electron position is known to be somewhere within the atom?

At first I tried to do it using ##\Delta x \Delta p\geq \frac{\hbar}{2}##
but then I found the expression ##\Delta k\Delta x\geq \frac{1}{2} ## in my text and so since uncertainty in position is equal to the radius of the atom (at least approximately) we have ##\Delta k \geq \frac{1}{2r_{atom}}## and therefore I just said the uncertainty in wavelength is just ##2r_{atom}##.

Q3: Does this model make sense? Explain.

Before I go trying to answer the last question and try to explain what I have done and found I'd like a little input on whether I am on the right track so far. Is my thinking correct here?

Thank you so much!
 
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  • #2
Q1. Eigenfunction of hydrogen atom is not an eigenfunction of momentum, therefore there must be an uncertainty in finding a given momentum value. Since the de Broglie's hypothesis reads as ##|\mathbf{p}|=h/\lambda##, the quantity of interest in this case should be ##|\mathbf{p}|##, or equivalently the kinetic energy. Again, the electron momentum in hydrogen atom has some uncertainty, therefore we can't really find an exact value, rather it's more practical to calculate the average/expected value. So, I would suggest you calculate the expectation value of kinetic energy ##\langle E_k \rangle##, and take its square root to find the "expected" momentum ##|\mathbf{p}|##, from this point it should be straightforward to associate to the "expected" wavelength.
PS: Words contained in double apostrophes "..." do not mathematically describe the actual averaged value.
The strange thing with your result there is that, as you go farther from the nucleus (##n## becomes bigger), the wavelength decreases which means the electron moves faster. This is of course counter-intuitive as when you go farther you should feel less electrostatic force from the nucleus.
 

Related to Ground state wavelength and uncertainty Bohr/deBroglie model

1. What is the ground state wavelength in the Bohr/deBroglie model?

In the Bohr/deBroglie model, the ground state wavelength is the shortest possible wavelength of an electron in the lowest energy level. It is calculated using the formula λ = h/mv, where h is Planck's constant, m is the mass of the electron, and v is the velocity of the electron.

2. How is the ground state wavelength related to the uncertainty principle?

The ground state wavelength is related to the uncertainty principle because it represents the minimum possible uncertainty in the position of an electron in the lowest energy level. This means that the exact location of the electron cannot be known, only the probability of finding it within a certain region.

3. Is the ground state wavelength the same for all atoms?

No, the ground state wavelength varies depending on the mass and charge of the atom. Heavier atoms will have a shorter ground state wavelength compared to lighter atoms.

4. How does the ground state wavelength change as the energy level increases?

As the energy level increases, the ground state wavelength becomes longer. This is because the electron's velocity increases as it moves to higher energy levels, resulting in a longer wavelength.

5. Can the ground state wavelength be measured experimentally?

Yes, the ground state wavelength can be measured experimentally using techniques such as spectroscopy. By analyzing the light emitted or absorbed by an atom, scientists can determine the energy levels and corresponding wavelengths of the electrons within the atom.

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