Probability of Ball Tunneling Through Wall at 1m/s

[Mr_Cloud]

Homework Statement

Suppose that a ball is tossed at a wall; what is
the probability that it will tunnel through to
the other side? The mass of the ball is 0.14 kg,
the width of the wall is a = 0.2 m, and the one
who tossed the ball was deadly tired, so that
the ball is tossed only weakly at 1.0 m/s.

a=0.2m
m=0.14kg
v=1m/s

Homework Equations

$$T=[1+\frac{V_{0}^{2}}{4E(V_{0}-E)}sinh^{2}(\frac{2a}{\hbar}\sqrt{2m(V_{0}-E)}]^{-1}$$

The Attempt at a Solution

I presume I must use that equation, though I have no idea what the value of V0 or (V0-E) is! How do I do this?

Thanks in advance for your help.

 

[mark726]

Hello,

Thank you for your question. In order to calculate the probability of the ball tunneling through the wall, we first need to understand the concept of tunneling. Tunneling is a quantum mechanical phenomenon where particles can pass through a potential barrier even if they do not have enough energy to overcome it. This is due to the uncertainty principle, which allows particles to have a small probability of existing on the other side of the barrier even if they do not have enough energy to reach it.

In this case, the ball is behaving like a quantum particle and we can use the Schrödinger equation to calculate the probability of it tunneling through the wall. The equation you have provided is known as the transmission coefficient, which gives us the probability of the particle tunneling through the barrier. However, this equation is typically used for electrons and not macroscopic objects like a ball.

To calculate the probability of the ball tunneling through the wall, we need to know the height of the potential barrier and the kinetic energy of the ball. In this case, the height of the barrier would be the height of the wall, and the kinetic energy of the ball can be calculated using the formula KE = 1/2 * m * v^2, where m is the mass of the ball and v is its velocity.

Once we have these values, we can plug them into the Schrödinger equation and calculate the transmission coefficient. However, the transmission coefficient only gives us the probability of the ball tunneling through the wall, it does not tell us whether it will actually happen or not.

I hope this helps to clarify the concept of tunneling and how it applies to this scenario. If you have any further questions, please let me know.