# How can Uncertainty principle be used to get speed of alpha particle

• strugglin
In summary, the uncertainty principle can be manipulated to estimate the speed of an alpha particle trapped in a nucleus, such as uranium 238. By using the Schrodinger equation, one can calculate the expectation values of momentum and speed. The uncertainty relation states that the ground state energy of the nucleus must be at least equal to the kinetic energy of the particle, and by solving for momentum and velocity, one can estimate the speed of the particle. This may require the use of relativity if the speed is close to the speed of light.
strugglin
assuming that Delta p = aprroxiamtely p, how can the uncertainty principle, be manipulated in order to achieve the speed of an alpha particle trapped in a nucleus of just say uranium 238 with proton number 92 ? or any other nucleus...what is the main manipulation steps of the formula?would appreciate any help, thanks!

for the uranium above I've estimated the size of the nucleus to have radius of around 7.44 fm or (7.44 x 10^-15) m if that's supposed to help, not sure

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I'm not sure how to solve this problem - just throwing some ideas out here.

Delta p is a little more than 'approximately p'. More formally:

Delta p is the root of { (the expectation value of (p squared) - (expectation value of p) squared }

i.e. {<p^2> - <p>^2} if you can read latex.

and similarly for Delta x.

So the uncertainty relation can be understood as relating these quantities in an inequality.

If you know the Schrodinger equation for the system you have in mind mention, you should be able to calculate some of these expectation values.

Presumably, when the questioner asks for the speed of an alpha particle, they're asking for the expectation value of the speed.

Speed is connected to momentum here.

This should at least give you something to play with - until somebody cleverer comes to help.

However there's one thing I'm very uncertain (haha) about - the uncertainty relation is merely an inequality. I've seen the uncertainty relation used to estimate properties of systems in their ground state - because we're searching for a minimal quantity - but this aspect doesn't seem to appear in the question. So I fear I'm missing something.

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First you need to estimate the ground state level of the nucleus. That is the kinetic energy, or p^2/2m. The ground state is stationary, so <p>=0. So using $$\Delta p=<(p-<p>)^2>^{1/2}$$, $$\Delta p=<p^2>^{1/2}=(2m<H>)^{1/2}$$.

So from the uncertainty relation $$\Delta p=(2m<H>)^{1/2}>\frac{\hbar}{2 \Delta x}$$

or $$<H> > \frac{\hbar^2}{8m (\Delta x)^2}$$

So what you know is that the potential has a value at least equal to this ground state energy, in order to be confined. So to escape the potential, the particle must have at least this energy. When it is free of the nucleus, it'll still have this energy relative to the nucleus. So at the very least, the kinetic energy is equal to $$\frac{\hbar^2}{8m (\Delta x)^2}$$, but it is actually more relative to infinity where it is free instead of relative to the nucleus. So set p^2/2m equal to $$\frac{\hbar^2}{8m (\Delta x)^2}$$, and solve for p:

$$p=\frac{\hbar}{2 (\Delta x)}$$

if the size of the nucleus is 10^-14, then the momentum is 3.3*10^(-20). If the alpha particle has mass 6.6*10^(-27), then the velocity is 1/2*10^7 m/s. I think this is close enough to the speed of light (if you had used 10^-15 meters for the nucleus instead of 10^-14 then you'd get there) where you might have to use relativity to get the speed, which means you'd set (c^2p^2+m^2c^4)^(1/2) equal to the energy, and solve for p, and then use the relativistic relationship between p and v to solve for v.

Anyways, if I didn't make a mistake, this is the long explanation of how to use the uncertainty principle.

## 1. What is the uncertainty principle and how does it relate to the speed of alpha particles?

The uncertainty principle, also known as Heisenberg's uncertainty principle, is a fundamental concept in quantum mechanics that states it is impossible to simultaneously know the exact position and momentum of a particle. This principle applies to all particles, including alpha particles, and therefore affects our ability to accurately measure their speed.

## 2. Can the uncertainty principle be used to directly calculate the speed of an alpha particle?

No, the uncertainty principle cannot be used to directly calculate the speed of an alpha particle. However, it can provide a range of possible speeds within which the particle is likely to be found.

## 3. How can the uncertainty principle be used to estimate the speed of an alpha particle?

By measuring the uncertainty in the position and momentum of an alpha particle, we can calculate a range of possible speeds. The smaller the uncertainty in position, the larger the uncertainty in momentum and vice versa. This means that a more precise measurement of one quantity will result in a less precise measurement of the other, making it difficult to accurately determine the speed of the alpha particle.

## 4. Are there any other factors that can affect the measurement of an alpha particle's speed?

Yes, there are other factors that can affect the measurement of an alpha particle's speed, such as the equipment used for measurement and any external forces acting on the particle. These factors can introduce additional uncertainties and errors into the measurement.

## 5. How can we minimize the uncertainty in measuring the speed of an alpha particle?

We can minimize the uncertainty in measuring the speed of an alpha particle by using more precise and advanced equipment, reducing external forces on the particle, and taking multiple measurements to obtain an average speed. Additionally, we can also use theoretical calculations and models to estimate the speed based on other known factors, such as the particle's energy and mass.

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