# Units of the given potential box

• phenomenologic
In summary, the problem involves a car with a mass of 1 ton driving into a barrier with a height of 1 meter and a thickness of 3 meters. The goal is to determine the probability of the car tunnelling through the barrier based on the given equation (i) for tunneling probability. The equation involves parameters such as the potential barrier, particle's energy, and barrier thickness. In order to calculate the energy parameter, it is suggested to use the units of Joules. However, this raises some issues and leads to conflicting results. One possible solution is to use the equation mgh = 1/2 mv^2 to calculate the energy of the particle.
phenomenologic

## Homework Statement

A car (particle) with mass m = 1 t drives into a barrier with a height of 1 m and a thickness of 3 m. The kinetic energy shall be sufficient to get classically over a barrier with half of the height. Derive an equation for the tunnelling probability. What is the probability to tunnel through the barrier?

## Homework Equations

Tunneling probability is given by

(i) T = (|A'|/|A|)^2 = (1−E/Vo)/[1 − E/Vo + (Vo/4E) sinh2(bα] ; where

α = √[2m(Vo − E)]/(h/2π) , b=3 m the barrier thickness, Vo: barrier potentail and E: particle's (car's) energy.

This is an experimental physics homework. So I should be getting some numbers at the end. I have the equation (i) and I know how to use it. But, I'm not sure what "the potential barrrier to having a height of 1 m" mean unit-wise. In other words, I don't know which units I should use for the energy while calculating α? Feels like I should be using Joules but it's a just an intiution. If so, why?

Apologies for not being able to use LaTeX.

Hello pheno,

In SI energy is in Joules. Doesn't give a very high speed, but I think that's what the composer of the exercise means.

So you suggest I use Vo = 1 J? If I do that, I will get

bα = b*√[2m(Vo − E)]/(h/2π) = 3√(2*1000*(1-1/2)/(1.054*10^-34)) ≈ 10^36. With Vo = 1 J, E = 0.5 J the equation (i) becomes

T=1/(sinh(bα))^2.

New problems arise:
1) WolframAlpha didn't calculate (sinh(10^36))^-2 so I did a Taylor expansion (just to get an idea about the value I should have):

(sinh(bα))^-2 = (bα + (bα)^3/3! +o(5))^-2 ≈ 10^-72 ≈ 0.

This could have been okay I guess, saying the probabilty of the car to tunnel is nearly zero.

2) However when I do the following to check my conclusion, I get something completely different.

For bα>10, we can approximate sinh(bα) = (1/2)(e^bα - e^-bα) ≈ (1/2)e^bα. Then 1/(sinh(bα))^2 ≈ 1/((1/4)e^(2bα)) = 4*e^(-2bα) = 4* e^(-2*10^36) ≈ 4.

This is clearly wrong, since T cannot be bigger then one. Hence, I'm stuck again.

Hint: mgh

## 1. What is a potential box and why is it important in scientific research?

A potential box is a theoretical construct used in quantum mechanics to describe the behavior of particles within a confined space. It is important in scientific research because it allows us to study the properties and behavior of particles in a controlled environment, which can help us understand the fundamental principles of quantum mechanics.

## 2. What are the units of measurement used in a potential box?

The units of measurement used in a potential box can vary depending on the specific system being studied. However, common units include meters (m) for length, joules (J) for energy, and kilograms (kg) for mass. These units are used to measure the position, energy, and mass of particles within the potential box.

## 3. How are the units of the potential box related to the size of the system?

The units of the potential box are directly related to the size of the system being studied. The length unit, for example, will be smaller if the potential box is smaller, and larger if the potential box is larger. Similarly, the energy unit will be smaller for a smaller potential box and larger for a larger potential box.

## 4. Can the units of the potential box be converted to other units?

Yes, the units of the potential box can be converted to other units using mathematical equations and conversion factors. However, it is important to note that the physical properties of the system will not change, only the units of measurement used to describe them.

## 5. How do the units of the potential box affect the behavior of particles within it?

The units of the potential box do not directly affect the behavior of particles within it. However, they can provide important information about the properties and behavior of particles, such as their energy levels and probability of being in a certain position. These factors, in turn, can influence the overall behavior of the system.

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