In quantum mechanics, the particle in a box model (also known as the infinite potential well or the infinite square well) describes a particle free to move in a small space surrounded by impenetrable barriers. The model is mainly used as a hypothetical example to illustrate the differences between classical and quantum systems. In classical systems, for example a ball trapped inside a heavy box, the particle can move at any speed within the box and it is no more likely to be found at one position than another. However, when the well becomes very narrow (on the scale of a few nanometers), quantum effects become important. The particle may only occupy certain positive energy levels. Likewise, it can never have zero energy, meaning that the particle can never "sit still". Additionally, it is more likely to be found at certain positions than at others, depending on its energy level. The particle may never be detected at certain positions, known as spatial nodes.

An illustration of Particle in a Box

(Downloaded from Wikipedia)

An illustration of Particle in a Box

(Downloaded from Wikipedia)

The formula to calculate the energy of a particle in a box is written as follows:

Where E is the total energy, n is an integer number (primary quantum number), h is the planck constant, m

Now my question is: Where did that equation come from?

Before we start to find it, I warn for those whom don't like calculus to not read this post. LOL.

Okay now, we start it from the Operator of Hamiltonian about Quantum Mechanics where:

Where T is the kinetic energy operator.

_{e}is the mass of particle, L is the length of the box.Now my question is: Where did that equation come from?

Before we start to find it, I warn for those whom don't like calculus to not read this post. LOL.

Okay now, we start it from the Operator of Hamiltonian about Quantum Mechanics where:

Where T is the kinetic energy operator.

And V is the Potential Energy Operator.

∇ (Read: Del) is the gradient operator where ∇

We already identify each operator. Now, we determine the meaning of The Hamiltonian Operator itself and Its relation with the Total Energy.

∇ (Read: Del) is the gradient operator where ∇

^{2}:We already identify each operator. Now, we determine the meaning of The Hamiltonian Operator itself and Its relation with the Total Energy.

Because in this case, we use one dimension only so that:

In the box, there's no potential energy because the velocity of particle is assumed as 0 so that:

Now let's play with this equation:

From here, we can find the characteristic value of the equation above (Don't ask me where does it come from, because I hate mathematics.)

Add the variables to the equation so that:

Now, let's find the value of ��.

Back to the picture above first before we continue this equation.

At the left barrier of the box the value of x = 0 and x = L for the right one. So that:

Thank You.

Source:

Documents of Pelatnas 2 IChO 2012

Wikipedia

Source:

Documents of Pelatnas 2 IChO 2012

Wikipedia

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