Quantum tunneling and radioactive decay.

[bayan]

Homework Statement

The edge of a nucleus can be roughly modeled as a square potential barrier. An alpha particle in an unstable nucleus can be modeled as a particle with a specific energy, bouncing back and forth between these square potential barrier.

Consider a nucleus of radius r and an alpha particle with kinetic energy E (i.e., let the potential energy within the nucleus be zero) and mass m.

Assuming that the alpha particle moves along a diameter of the nucleus and that it moves at low enough speed that relativistic effects are negligible, what is the time tau between successive encounters between each edge of the nucleus and the alpha particle?

Express your answer in terms of $K_{e}$, $r$, and $m$.

Homework Equations

$K_{e}=mv^{2}$

The Attempt at a Solution

$v^{2}=\frac{(2r)^{2}}{t^{2}}$

$\frac{K_{e}}{m}=\frac{(2r)^{2}}{t^2}$

$\sqrt{\frac{K_{e}}{m}}=\frac{2r}{t}$

$2r\sqrt{\frac{m}{K_{e}}}={t}$

Have I used the right approach to this problem? and have I got the correct answer?

Thanks in advance

 

[bayan]

Seems like the answer does not need the 2 in my answer, which means I have made a mistake in my approach or calculation. Anyone able to help out?

Thanks.

 

[Curious3141]

I'm not really sure about your approach, but there is a factor of half missing in your KE equation:

$$K_e = \frac{1}{2}mv^2$$

Does that help?

 

[bayan]

I'm not really sure about your approach, but there is a factor of half missing in your KE equation:

$$K_e = \frac{1}{2}mv^2$$

Does that help?

I should hope so, lol

Thanks :)

 

[asteriatic]

I cannot confirm if your approach is correct without seeing your calculations and assumptions. However, the general concept of using the kinetic energy and mass of the alpha particle to determine the time between successive encounters seems reasonable. Additionally, the use of the square potential barrier model is a common approach in studying quantum tunneling and radioactive decay.

To confirm the accuracy of your answer, you can compare it with experimental data or use it to make predictions in other scenarios. It may also be helpful to consult with a colleague or mentor to review your approach and calculations. Overall, as long as you have a logical and well-supported solution, your answer can be considered valid.