Probability of electron in a box problem

• TLeo198
In summary, the conversation discusses calculating the probability of finding an electron in certain intervals when its wavefunction is given. The conversation also mentions the normalization equation and the principle of the probabilistic interpretation of quantum mechanics. The conversation ends with a request for help in constructing a new function that is orthogonal and normalized.
TLeo198

Homework Statement

Calculate the probability that an electron will be found (a) between x = 0.1 and 0.2 nm, (b) between 4.9 and 5.2 nm in a box of length L = 10 nm when its wavefunction is psi = (2/L)^1/2 sin(2*pi*x/L).

Homework Equations

As far as I know, only the normalization equation, which is A^2 * Integral(psi^2) = 1

The Attempt at a Solution

My problem is actually how to start it. Should I assume that the wavefunction is already normalized, or should I normalize the wavefunction by equating the integral of the squared wavefunction to 1, then solve for A? Sorry if wording of the question is bad. Any help is appreciated.

It looks like the wavefunction is already normalized. Its easy to check.

The fundamental principle of the probabilistic interpretation of quantum mechanics is that the probability of finding a particle between points a and b, is equal to the integral of the wave-function squared over that interval:

$$P(a<x<b) = \int_a^b |\Psi(x) |^2 dx$$

I was able to solve it, thanks!

Given two normalized but non-orthogonal eigen functions, psi=(1/sqrt(pi))exp(-r) and phi=(1/sqrt(3pi))rexp(-r). Construct a new function PSI which is orthogonal to the first function and is normalized.

Many many thanks for your kind help.

MSPNS

I would approach this problem by first understanding the concept of probability in quantum mechanics. In quantum mechanics, the probability of finding a particle at a certain position is given by the squared magnitude of its wavefunction at that position. Therefore, to calculate the probability of finding an electron in a certain range of positions, we need to integrate the squared wavefunction over that range.

For part (a), we need to integrate the squared wavefunction from x = 0.1 nm to x = 0.2 nm. This can be done by breaking up the integral into two parts, from x = 0.1 nm to x = 0.2 nm and from x = 0 to x = 0.1 nm, and then subtracting the second part from the first. This is because the wavefunction is zero for x < 0 nm.

The integral for the first part would be A^2 * Integral(sin^2(2*pi*x/L)) from x = 0.1 nm to x = 0.2 nm. This can be simplified using trigonometric identities to give A^2 * (0.1 - 0.2 + (sin(4*pi/10) - sin(8*pi/10)) / (4*pi)). Similarly, the integral for the second part would be A^2 * Integral(sin^2(2*pi*x/L)) from x = 0 to x = 0.1 nm, which simplifies to A^2 * (sin(2*pi/10) - sin(8*pi/10)) / (4*pi).

Subtracting the second part from the first, we get A^2 * (0.1 - (sin(2*pi/10) - sin(8*pi/10)) / (4*pi)). Since we know that the integral of the squared wavefunction over all space should be equal to 1, we can set this expression equal to 1 and solve for A. This gives us A = (2*pi)^1/2.

We can then use this value of A to calculate the probability of finding the electron between x = 0.1 and 0.2 nm by plugging it into the integral expression and evaluating it. This gives us a probability of approximately 0.077.

For part (b), the process is similar. We need to integrate the squared wavefunction from x = 4.9 nm to

1. What is the probability of finding an electron in a specific region of the box?

The probability of finding an electron in a specific region of the box is determined by the wave function of the electron. The square of the wave function, known as the probability density, gives the probability of finding the electron at a particular location in the box.

2. How does the size of the box affect the probability of finding an electron?

The size of the box directly affects the probability of finding an electron. As the box size decreases, the energy levels of the electron become more closely spaced, resulting in a higher probability of finding the electron at any given location in the box.

3. What is the significance of the boundary conditions in the probability of electron in a box problem?

The boundary conditions, such as the requirement for the wave function to be continuous and zero at the edges of the box, play a crucial role in determining the allowed energy levels and the corresponding probabilities of finding an electron in the box. These conditions ensure that the wave function is physically meaningful and that the electron is confined within the box.

4. How does the number of energy states in the box affect the probability of finding an electron?

The number of energy states in the box is directly proportional to the probability of finding an electron. As the number of energy states increases, the probability of finding the electron at a particular energy level or location in the box also increases.

5. Can the probability of finding an electron in a box be greater than 1?

No, the probability of finding an electron in a box cannot be greater than 1. This is because the total probability of finding the electron at any location in the box must equal 1, as the electron must be somewhere in the box at all times due to the boundary conditions.

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