The electron is in a potential box of width l.

There is an electron in a potential box of width l. It is necessary to find points at which it can be in the interval (0,l). The solution to problem 60307 is presented below.

Let us write the Schrödinger equation for the one-dimensional case:

HΨ=EΨ

where H is the Hamilton operator, Ψ is the wave function, E is the electron energy.

In a potential box, the potential energy is zero, i.e. V(x)=0. Then the Hamilton operator has the form:

H=-(h^2/2m) * d^2/dx^2

where h is Planck's constant, m is the electron mass.

We are looking for a solution to the Schrödinger equation in the form:

Ψ(x)=Asin(kx)+Bcos(kx)

where A and B are constants, k=sqrt(2mE)/h.

The conditions for an electron to be in a potential box are:

• Ψ(0)=0
• Ψ(l)=0

From the first condition it follows that B=0, and from the second condition it follows that k=nπ/l, where n is an integer.

Then the equation for the wave function will take the form:

Ψ(x)=Asin(nπx/l)

Let's find the constant A. Using the normalization condition, we get:

∫Ψ^2(x)dx=1

∫Asin^2(nπx/l)dx=1

A^2*l/2=1

A=sqrt(2/l)

Thus, the wave function of an electron in a potential box of width l has the form:

Ψn(x)=sqrt(2/l)*sin(nπx/l)

where n is an integer.

The points at which an electron can be in the interval (0,l) are determined by the conditions:

• Ψn(0)≠0
• Ψn(l)≠0

Taking into account the form of the wave function, we obtain:

• Ψn(0)=sqrt(2/l)*sin(0)=0
• Ψn(l)=sqrt(2/l)*sin(nπ)=0 and n=1,2,3,.

Thus, the electron can be located at the points:

x=k*l/(2n+1), where k=1,2,3,... and n=0,1,2,3,...

If you have any questions regarding the solution, please write. I will try to help.

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Product description: "?electron in a potential box"

We offer you a unique digital product - “?electron in a potential box”. This product contains a detailed description of the properties and behavior of an electron in a potential box of width l.

You will learn the points at which an electron can be located in the interval (0, l), as well as the conditions that the wave function of an electron in a potential box must satisfy. All information is presented in an accessible format, using formulas and laws explained with examples.

The product has a beautiful html design, which makes it easy and quick to familiarize yourself with the material, while maintaining its structure and logical sequence.

By purchasing this product, you will expand your knowledge in the field of physics and gain a complete understanding of the properties of the electron in a potential box.

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