Finding the length of a box confining an electron?

In summary, the problem involves finding the length of the smallest box that can confine an electron while keeping its speed below 66 m/s. This can be solved using the Heisenberg uncertainty principle, where one of the approximate forms involves the product of the uncertainties in position and momentum being greater than or equal to h. The specific form to be used may vary depending on the course or textbook.
  • #1
HenryHH
12
0

Homework Statement



You want to confine an electron and you want to know for certain that the electron's speed is no
more than 66 m/s. What is the length of the smallest box in which you can do this?

A) 2.8 × 10^-6m B) 1.4 × 10^-6m C) 1.1 × 10^-5m D) 5.5 × 10^-6m


Homework Equations



There are two equations that could be used: E = (1/2m)(hn/2L)^2 or E = (h^2/8mL^2)(n^2), where n = 1, 2, 3, 4...

The Attempt at a Solution



I understand that I will be solving for L. However, I don't understand how v = 66 m/s will plug into either of the equations above. Also, I'm not sure how to know what to plug in for n. I know the answer is D.) 5.5 x 10^-6m, but I have no idea how they got that answer. Assistance would be greatly appreciated.
 
Physics news on Phys.org
  • #2
I don't know, the question is weird to me. Without having really worked out the problem myself, I'd just try nonrelativistic kinetic energy relation because it will be quick and easy. If that doesn't work you may have to use a relativistic momentum energy relation.
 
  • #3
I think this is just a simple application of the Heisenberg uncertainty principle.

There are multiple ways to express the Heisenberg uncertainty principle. One way is in terms of an approximate formula involving uncertainties in terms of "deltas." Another form is a very precise inequality that involves standard deviations, and its use generally requires accurate information about the specific wavefunction shape.

I'm guessing this problem involves the much easier approximation with the "delta" uncertainties. If you use that one, the answer is one of the listed choices.

--------------
Edit: But again, there are different ways to express the Heisenberg uncertainty principle. In its simplest form,

[tex] \Delta p \Delta x \approx h [/tex]

But then there is a slightly more approximate inequality,

[tex] \Delta p \Delta x \gtrsim h [/tex]

And to make things more confusing, that's sometimes expressed by

[tex] \Delta p \Delta x \gtrsim \frac{h}{2} [/tex]

But the most exact version requires you know quite a bit about the wavefunction. You can calculate the variance of the position and variance of the momentum using standard quantum mechanical operators. For example, assuming the expectation value of position and momentum are both 0, then in 1-demension,
[tex] \sigma_x^2 = \int_{-\infty}^{\infty} \psi^* x^2 \psi \ dx [/tex]
[tex] \sigma_p^2 = \int_{-\infty}^{\infty} \psi^* \left( -\hbar^2 \frac{\partial^2}{\partial x^2} \right) \psi \ dx [/tex]
Then in terms of standard deviations, one can show the ultimate in the uncertainty principle:
[tex] \sigma_x \sigma_p \geq \frac{\hbar}{2} [/tex]
where [itex] \hbar [/itex] = h/(2π)

I'm guessing the version you are supposed to use is one of the first three. Check your textbook/coursework for the preferred version in your course.
 
Last edited:
  • #4
Oh yeah, I think collins is right now that I take a second read. The key wording I missed was "no more," which I guess the problem author thinks can be interpreted as "within" or even (gasp) "uncertain." Sorry for ranting, but if a problem has to resort to language to be hard it's a bad problem.
 
  • #5


I would approach this problem by first understanding the concept of confinement and its relationship to the energy of an electron. In this case, the electron is confined in a box with a certain length, and this confinement affects the electron's energy.

The equations provided in the problem are known as the energy quantization equations for a particle confined in a box. These equations relate the energy of the particle (in this case, the electron) to its mass, the length of the box, and a quantum number (n) which determines the energy level of the particle.

To solve for the length of the box, we need to use the equation E = (h^2/8mL^2)(n^2), where h is Planck's constant (6.626 x 10^-34 J*s), m is the mass of the electron (9.11 x 10^-31 kg), and n is the quantum number. We also know that the maximum speed of the electron is 66 m/s.

To confine the electron in the box with a maximum speed of 66 m/s, we need to ensure that its energy is less than or equal to the maximum kinetic energy it can have at this speed. This means that we can set the energy (E) equal to the maximum kinetic energy, which is given by the equation E = (1/2)m(v^2), where m is the mass of the electron and v is its speed.

Plugging in the known values, we get:

(1/2)m(v^2) = (h^2/8mL^2)(n^2)

Solving for L, we get:

L = √[(h^2/4mv^2)(n^2)]

Now, we need to find the minimum value of L that will satisfy the condition that the electron's speed is no more than 66 m/s. This means that we need to find the minimum value of n that will give us a value of L that is greater than or equal to the length of the box we are looking for.

To do this, we can start with n = 1 and plug in the values for h, m, and v to calculate the corresponding value of L. If this value is greater than or equal to the lengths given in the answer choices, then we have our answer. If not, we can increase the value of n and repeat the process until we find the
 

1. How do you find the length of a box confining an electron?

The length of a box confining an electron can be found by using the Schrödinger equation, which describes the behavior of quantum particles such as electrons. This equation takes into account the size of the box as well as the energy and mass of the electron to determine the length.

2. What is the significance of finding the length of a box confining an electron?

Finding the length of a box confining an electron is important because it allows us to understand the behavior of electrons in confined spaces, which is essential for many technological applications such as transistors and microchips. It also helps us to understand the fundamental principles of quantum mechanics.

3. Can the length of a box confining an electron be measured directly?

No, the length of a box confining an electron cannot be measured directly. This is because the concept of length in quantum mechanics is different from classical mechanics, and the uncertainty principle states that it is impossible to know both the position and momentum of a particle precisely at the same time.

4. How does the length of a box affect the behavior of an electron?

The length of a box affects the behavior of an electron by limiting its movement and confining it to a specific space. This confinement leads to quantization of energy levels, meaning the electron can only have certain discrete energy values within the box. The size of the box also affects the probability of finding the electron in a certain location within the box.

5. Are there any factors that can influence the length of a box confining an electron?

Yes, there are several factors that can influence the length of a box confining an electron. These include the energy and mass of the electron, the potential energy of the box, and the material properties of the box itself. Additionally, external forces such as electric or magnetic fields can also affect the length of the box and therefore the behavior of the electron.

Similar threads

  • Introductory Physics Homework Help
Replies
30
Views
5K
  • Introductory Physics Homework Help
Replies
1
Views
3K
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
1
Views
751
  • Introductory Physics Homework Help
Replies
12
Views
1K
Replies
6
Views
684
  • Introductory Physics Homework Help
Replies
3
Views
735
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
1K
  • Introductory Physics Homework Help
Replies
2
Views
3K
Back
Top