Chapter 41: Problem 40
An electron is in a three-dimensional box with side lengths \(L_X =\) 0.600 nm and \(L_Y = L_Z = 2L_X\). What are the quantum numbers \(n_X, n_Y,\) and \(n_Z\) and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy due to spin)?
Short Answer
Expert verified
Quantum numbers for lowest levels are: (1,1,1), (1,1,2), (1,2,1), (2,1,1). Energies are computed using energy formula. Degeneracies include spin.
Step by step solution
01
Understand the Problem
We have an electron in a 3D box with sides \( L_X = 0.600 \text{ nm} \) and \( L_Y = L_Z = 2L_X \). We need to find possible quantum numbers \( n_X, n_Y, \) and \( n_Z \) for the four lowest energies and also calculate these energies.
02
Write the Formula for Energy Levels in a 3D Box
The energy levels for an electron in a 3D box are given by the formula: \[ E_{n_x,n_y,n_z} = \left(\frac{h^2}{8m}\right) \left(\frac{n_x^2}{L_X^2} + \frac{n_y^2}{L_Y^2} + \frac{n_z^2}{L_Z^2}\right) \] where \( h \) is Planck's constant, \( m \) is the electron mass, and \( n_X, n_Y, n_Z \) are the quantum numbers.
03
Substitute Values into the Energy Formula
Convert \( L_Y \) and \( L_Z \) to lengths in nm: \( L_Y = L_Z = 1.2 \text{ nm} \). Substitute \( L_X = 0.600 \text{ nm} \) and \( L_Y = L_Z = 1.2 \text{ nm} \) into the energy formula.
04
Determine Quantum Numbers for the Four Lowest Energy Levels
For the lowest energy states, try combinations of \( n_X, n_Y, \) and \( n_Z \) where the sum of their squares divided by the respective box lengths yields the smallest values. The lowest four combinations are (1,1,1), (1,1,2), (1,2,1), and (2,1,1).
05
Calculate the Energies
Compute the energies using the formula. For the first combination (1,1,1), compute as follows: \[ E_{1,1,1} = \left(\frac{h^2}{8m}\right) \left(\frac{1^2}{(0.600 \text{ nm})^2} + \frac{1^2}{(1.2 \text{ nm})^2} + \frac{1^2}{(1.2 \text{ nm})^2} \right) \] Repeat for the other combinations (for \( n_X, n_Y, n_Z\) as (1,1,2), (1,2,1), (2,1,1)). Calculate energies in eV.
06
Calculate Degeneracy Including Spin
Since the electron can have spin up or spin down, there is a degeneracy of 2 for each non-degenerate state. For states with identical energy levels like (1,2,1), (2,1,1), and (1,1,2), add their degeneracies. Calculate total degeneracy for each level.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Three-dimensional box
A three-dimensional box in quantum mechanics is a simple theoretical model used to understand the behavior of particles like electrons confined in a compact space. Imagine having a box where an electron moves freely within certain boundaries. These boundaries in our case are defined by the side lengths function:
- Two sides, \(L_Y\) and \(L_Z\), are doubled in length compared to one side, \(L_X\), implying the box isn't cubical.
- In our specific example, \(L_X\) is 0.600 nm, meaning \(L_Y\) and \(L_Z\) are each 1.2 nm (since \(L_Y = L_Z = 2L_X\)).
Quantum numbers
Quantum numbers are essential tools in quantum mechanics. They help us pinpoint the allowed energy levels for particles within confined systems like our three-dimensional box. Each quantum number represents one dimension of the box and how the particle's position or state is quantized within that dimension.
- \(n_X\), \(n_Y\), and \(n_Z\) represent quantum numbers in each of the three dimensions respectively.
- These numbers are integers starting from 1 (not zero). They designate the node counts in standing wave patterns in the box.
Energy levels
Energy levels in a quantum box are derived based on a mathematical model. The energy at each level can be mathematically described as: \[E_{n_X,n_Y,n_Z} = \left(\frac{h^2}{8m}\right) \left(\frac{n_X^2}{L_X^2} + \frac{n_Y^2}{L_Y^2} + \frac{n_Z^2}{L_Z^2}\right)\]where:
- \(E_{n_X,n_Y,n_Z}\) represents the energy corresponding to quantum numbers \(n_X, n_Y,\) and \(n_Z\).
- \(h\) is Planck's constant and \(m\) is the mass of the electron.
Electron spin
Electron spin is an intrinsic form of angular momentum carried by electrons. It can have one of two possible values:
- Spin-up (+1/2)
- Spin-down (-1/2)
- Each energy state is doubled in possibility due to the two independent spin states, leading to an effective degeneracy of 2.