A particle of mass 2.00×10^−10 kg is confined in a hollow cubical three-dimensional box, each edge of which has a length, 2.00×10^−10 m, and for which the potential energy function is zero inside, and infinite outside, the box. The total energy of the particle is 2.47×10^−37 J. What are the quantum numbers that correspond to each of the three possible quantum states?
(b) (i) If the same particle is instead confined in one dimension between infinitely high potential barriers, with the same energy and in the same size region as above, what is the single quantum number that characterises the wavefunction of the particle when it occupies this energy level?
(ii) Explain why the particle is more likely to be detected in a small region centred on a position 0.50×10^−10 m from either wall than in a small region centred on a position 1.00×10^−10 m from either wall.
I hesitate to reply because I have not done this sort of problem since 1958. However look at:
http://phy240.ahepl.org/Serway-9-QM-in-3D.pdf
For the three-dimensional box, the total energy of the particle is given by:
E = (n₁² + n₂² + n₃²) * (h²/8mL²)
where n₁, n₂, and n₃ are the quantum numbers for the x, y, and z directions respectively, h is Planck's constant, m is the mass of the particle, and L is the length of each edge of the box.
Given:
E = 2.47×10⁻³⁷ J
m = 2.00×10⁻²⁰ kg
L = 2.00×10⁻¹⁰ m
Substituting these values, we can solve for the quantum numbers:
2.47×10⁻³⁷ J = (n₁² + n₂² + n₃²) * [(6.63×10⁻³⁴ J s)² / (8 * (2.00×10⁻²⁰ kg) * (2.00×10⁻¹⁰ m)²)]
Simplifying the equation:
2.47×10⁻³⁷ J = (n₁² + n₂² + n₃²) * 5.46×10²⁴ J s² kg⁻¹ m⁻²
Now, we can solve for the quantum numbers:
(n₁² + n₂² + n₃²) = (2.47×10⁻³⁷ J) / (5.46×10²⁴ J s² kg⁻¹ m⁻²)
(n₁² + n₂² + n₃²) ≈ 4.525×10⁻¹⁴ kg m³ s⁻²
We can make some observations from the equation above:
1. The values of the quantum numbers (n₁, n₂, n₃) must be integers or whole numbers.
2. The sum of their squares (n₁² + n₂² + n₃²) must be approximately equal to 4.525×10⁻¹⁴ kg m³ s⁻².
For the one-dimensional box, the total energy is given by:
E = (n² * h²) / (8mL²)
where n is the quantum number for the 1D box.
Given:
E = 2.47×10⁻³⁷ J
m = 2.00×10⁻²⁰ kg
L = 2.00×10⁻¹⁰ m
Substituting these values, we can solve for the quantum number:
2.47×10⁻³⁷ J = (n² * [(6.63×10⁻³⁴ J s)²]) / (8 * (2.00×10⁻²⁰ kg) * (2.00×10⁻¹⁰ m)²)
Simplifying the equation:
2.47×10⁻³⁷ J = (n² * 5.46×10²⁴ J s² kg⁻¹ m⁻²)
Now, we can solve for the quantum number:
n² = (2.47×10⁻³⁷ J) / (5.46×10²⁴ J s² kg⁻¹ m⁻²)
n² ≈ 9.038x10⁻¹³ kg m³ s⁻²
We can make some observations from the equation above:
1. The value of the quantum number (n) must be an integer or a whole number.
2. The square of the quantum number (n²) must be approximately equal to 9.038x10⁻¹³ kg m³ s⁻².
Now, let's address part (ii) of the question:
The particle is more likely to be detected in a small region centered on a position 0.50×10⁻¹⁰ m from either wall than in a small region centered on a position 1.00×10⁻¹⁰ m from either wall because the wavefunction of the particle is more concentrated around the center of the box. The probability density of finding the particle is higher where the wavefunction is more concentrated. In the case of being detected in a small region centered on a position closer to the wall, the wavefunction has a lower amplitude and is spread out over a larger space, resulting in a lower probability density.
To find the quantum numbers for the particles in the three-dimensional box, you need to solve the Schrödinger equation for a particle in a box. The Schrödinger equation for a particle of mass m in a one-dimensional box is given by:
-d^2ψ(x)/dx^2 + V(x)ψ(x) = Eψ(x)
Where ψ(x) is the wavefunction, V(x) is the potential energy function, E is the total energy, and x is the position.
For a three-dimensional box, you can similarly solve the Schrödinger equation by considering the wavefunction as a product of three independent one-dimensional wavefunctions, each representing a different dimension. The potential energy function is zero inside the box and infinite outside, indicating that the particle is confined within the box.
The energy levels of a particle in a box are given by:
E = (π^2ħ^2/2m)L^2 * (n_x^2 + n_y^2 + n_z^2)
Where ħ is the reduced Planck's constant, m is the mass of the particle, and L is the length of each edge of the box. n_x, n_y, and n_z are the quantum numbers corresponding to the different dimensions.
We are given the total energy E = 2.47×10^-37 J and the mass m = 2.00×10^-10 kg. With a little bit of manipulation, you can solve for the quantum numbers:
(n_x^2 + n_y^2 + n_z^2) = E / [(π^2ħ^2/2m)L^2]
Now, substitute the given values and calculate the right-hand side. This will give you the value of (n_x^2 + n_y^2 + n_z^2). Then you need to consider all possible combinations of quantum numbers that add up to this value.
For example, if (n_x^2 + n_y^2 + n_z^2) = 3, the possible combinations are (1,1,1), (3,0,0), (0,3,0), (0,0,3), (1,1,1), (2,1,0), (2,0,1), (1,2,0), (1,0,2), (0,2,1), (0,1,2).
Now, moving on to part (b):
(i) If the particle is confined in one dimension between infinitely high potential barriers, the wavefunction is characterized by a single quantum number. In this case, the energy levels are given by:
E = (π^2ħ^2/2m)L^2 * n^2
Where n is the single quantum number.
To find the value of n, substitute the given values into the equation and solve for n:
n^2 = E / [(π^2ħ^2/2m)L^2]
(ii) Now, let's consider why the particle is more likely to be detected in a small region centered on a position 0.50×10^-10 m from either wall than in a small region centered on a position 1.00×10^-10 m from either wall.
In a one-dimensional box, the probability density function (|ψ(x)|^2) describes the likelihood of finding the particle at a specific position. The probability density is higher in regions where the wavefunction is larger in magnitude.
In this case, the wavefunction for the particle confined in one dimension would have a sinusoidal shape between the potential barriers. The magnitude of the wavefunction is highest near the center of the box and decreases towards the walls. This means that the particle is more likely to be detected near the center of the box. So, a small region centered on a position 0.50×10^-10 m from either wall would have a higher probability of detecting the particle compared to a small region centered on a position 1.00×10^-10 m from either wall.