# Minimum possible energy for a particle in a box

• cojewmaw
In summary, the conversation discusses finding the lowest possible energy for a particle trapped in a box of size 1. This is done by considering the expected energy for a particle with zero momentum, but with minimal error. The formula E=p^2/2m can be used, replacing p with dp, to approximate the minimum energy. However, solving Schrödinger's equation is necessary for an exact calculation.
cojewmaw
I have a homework problem that is giving me some problems.

Consider a particle trapped in a box of size 1. All you know is that the particle is in the box. From that you can find what is called the lowest possible energy for that particle. What you really find is the energy expected for a particle whose momentum is zero but with som minimal error bar. If you say that p=0 +/- delta p, then you're really saying that the particle might as well have p=delta p. From that you can find the minimum possible energy of a particle in a box (and note that it's not 0!) (not 0 factoral, that's 1).

So I need to find the lowest possible energy. I really don't know where to start, any help would be very useful!

You could use the formula E=p^2/2m (good for a state with clearly defined p) and replace p with dp (since this is a typical value of p, if your distribution is centered at p=0). You can get minimum value for dp from uncertainity principle. But you must know that this is only an aproximation: exact calculation involves solving Schrödinger's equation.

I can provide you with some guidance on how to approach this problem. The concept of a particle in a box is a fundamental concept in quantum mechanics, where a particle is confined to a small region and its energy is constrained by the boundaries of the box. In this case, the box has a size of 1, meaning that the particle can only exist within this region.

To find the minimum possible energy for this particle, we need to use the Schrödinger equation, which describes the behavior of quantum particles. This equation includes the Hamiltonian operator, which represents the total energy of the system. In the case of a particle in a box, the Hamiltonian operator contains the kinetic energy term, which is related to the particle's momentum.

To solve this problem, you will need to set up the Schrödinger equation for a particle in a box and solve for the lowest energy state. This can be done by finding the eigenvalues and eigenvectors of the Hamiltonian operator.

Alternatively, you can also use the uncertainty principle, which states that the more precisely we know a particle's position, the less we know about its momentum and vice versa. In this case, since we know the particle is confined to the box of size 1, we can use this information to determine the minimum possible energy of the particle.

In summary, to find the minimum possible energy for a particle in a box, you will need to use the Schrödinger equation or the uncertainty principle. I recommend consulting your textbook or seeking help from your instructor or classmates to better understand these concepts and successfully solve the problem. Good luck!

## 1. What is the "particle in a box" concept in physics?

The "particle in a box" concept in physics refers to a theoretical model in which a particle is confined to a one-dimensional box with impenetrable walls. This model is often used to study the behavior of quantum particles, such as electrons, in a confined space.

## 2. What is the minimum possible energy for a particle in a box?

The minimum possible energy for a particle in a box is known as the ground state energy. This is the lowest energy level that the particle can have while still remaining in the box. It is determined by the size of the box and the mass of the particle.

## 3. How is the minimum possible energy for a particle in a box calculated?

The minimum possible energy for a particle in a box is calculated using the Schrödinger equation, a fundamental equation in quantum mechanics. This equation takes into account the size of the box, the mass of the particle, and the potential energy of the particle within the box.

## 4. What is the significance of the minimum possible energy for a particle in a box?

The minimum possible energy for a particle in a box is significant because it helps us understand the behavior of quantum particles in confined spaces. It also serves as a reference point for calculating the energy levels and probabilities of a particle in a more complex system.

## 5. Can the minimum possible energy for a particle in a box ever be exceeded?

No, the minimum possible energy for a particle in a box cannot be exceeded. This is because the walls of the box are impenetrable, meaning the particle cannot escape and therefore cannot have an energy level higher than the ground state energy. However, the particle can have different energy levels within the box, but the ground state energy will always be the lowest possible energy level.

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