Minimum possible energy for a particle in a box

[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!

 

[Lojzek]

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.

 

[denian]

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!