Consider a single electron in a 1dimensional box with size a,
What is the expected position <x> of the electron at the second excited state?
What is the expected value of linear momentum at the ground state ?
To determine the expected position of the electron at the second excited state in a 1-dimensional box with size a, we need to use the concepts of quantum mechanics and wave functions.
In this system, the electron is confined to a 1-dimensional box, meaning that it can only move along a line between two points. The size of the box is given by the parameter 'a'.
The wave function for the electron in the n-th excited state of a 1-dimensional box is given by:
ψ(x) = √(2/a) * sin((n * π * x) / a)
where x represents the position of the electron within the box, and n represents the quantum number of the excited state (n = 1, 2, 3, ...).
To find the expected position <x> of the electron at the second excited state (n = 2), we need to find the average value of x for the wave function ψ(x) associated with that state.
The expected position is given by the integral:
<x> = ∫ (-∞ to +∞) x |ψ(x)|^2 dx
where |ψ(x)|^2 represents the probability density function of the electron within the box.
Substituting the expression for ψ(x) and evaluating the integral over the limits of the box (0 to a), we get:
<x> = ∫ (0 to a) x |ψ(x)|^2 dx
= ∫ (0 to a) x * [(2/a) * sin^2((2 * π * x) / a)] dx
To solve this integral, we can use the trigonometric identity: sin^2θ = (1 - cos(2θ))/2
<x> = (2/a) * ∫ (0 to a) x * [(1 - cos((4 * π * x) / a))/2] dx
After evaluating this integral, you will get the expected position <x> of the electron in the second excited state.
Now, let's move on to the expected value of linear momentum at the ground state (n = 1). The linear momentum is given by:
p = -iħ(dψ/dx)
where ħ is the reduced Planck's constant.
To find the expected value of linear momentum in the ground state, we need to calculate the average value of p using the wave function ψ(x) associated with that state.
The expected value of linear momentum is given by the integral:
<p> = ∫ (-∞ to +∞) -iħ(dψ/dx) * ψ* dx
where ψ* represents the complex conjugate of the wave function.
Substituting the expression for ψ(x) and evaluating the integral over the limits of the box (0 to a), we get:
<p> = ∫ (0 to a) -iħ(dψ/dx) * ψ* dx
= ∫ (0 to a) -iħ * [(2/a) * sin((π * x) / a) * sin((π * x) / a)] * ψ* dx
To solve this integral, we can differentiate ψ(x) with respect to x and express it in terms of ψ(x). Then, we can calculate the expression for <p>.
After evaluating this integral, you will get the expected value of linear momentum at the ground state.