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Chapter 40: Problem 21

Seven electrons are trapped in a one-dimensional infinite potential well of width \(L .\) What multiple of \(h^{2} / 8 m L^{2}\) gives the energy of the ground state of this system? Assume that the electrons do not interact with one another, and do not neglect spin.

Short Answer

Expert verified
58 times \(\frac{h^2}{8mL^2}\).

Step by step solution

01

Understand the Problem

This problem involves determining the ground state energy for a system of seven non-interacting electrons in a one-dimensional infinite potential well of width \(L\). Each electron has a spin which is an important factor to consider as two electrons can occupy the same energy level if they have opposite spins.
02

Calculate Energy Levels for a Single Electron

The energy levels for a particle in a one-dimensional infinite potential well are given by\[E_n = \frac{h^2}{8mL^2} n^2,\]where \(n\) is a quantum number. Each electron can occupy these discrete energy levels inside the well.
03

Assign Electrons to Energy Levels

For seven electrons, we must consider Pauli's exclusion principle and the fact that each energy level can be occupied by two electrons of opposite spin. The distribution is as follows: 2 electrons in \(n=1\), 2 electrons in \(n=2\), 2 electrons in \(n=3\), and the last electron in \(n=4\).
04

Calculate Ground State Energy

The ground state energy is the sum of the energies for the occupied levels:\[E = 2\left(\frac{h^2}{8mL^2} \cdot 1^2\right) + 2\left(\frac{h^2}{8mL^2} \cdot 2^2\right) + 2\left(\frac{h^2}{8mL^2} \cdot 3^2\right) + 1\left(\frac{h^2}{8mL^2} \cdot 4^2\right).\]Simplifying this sum gives:\[E = \frac{h^2}{8mL^2} (2 \times 1 + 2 \times 4 + 2 \times 9 + 16) = \frac{h^2}{8mL^2} \cdot 58.\]
05

Conclusion

The total ground state energy is 58 times \(\frac{h^2}{8mL^2}\). Therefore, the multiple of \(\frac{h^2}{8mL^2}\) that gives the energy of the ground state of this electron system is 58.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Potential Well
An infinite potential well, also known as an infinite square well, is a fundamental concept in quantum mechanics. It represents a hypothetical box with infinitely high walls, meaning that a particle trapped inside cannot escape. In one dimension, the well has width \(L\) and perfectly rigid boundaries. As a result, the particle can only occupy certain discrete energy levels.
  • These energy levels are solutions to the Schr枚dinger equation within the potential well.
  • They depend on the quantum number \(n\), which determines the energy quantization.
  • Mathematically, the energy levels \(E_n\) are given by: \(E_n = \frac{h^2}{8mL^2} n^2\).
Understanding this model helps illustrate the concept of quantization in confined systems, an essential principle in quantum mechanics.
Pauli Exclusion Principle
The Pauli Exclusion Principle is a critical rule in quantum mechanics, particularly in dealing with multi-electron systems. It states that no two fermions, which include electrons, can occupy the same quantum state simultaneously within a quantum system.
  • This principle is responsible for the distinct configuration of electrons in atoms and molecules.
  • In an infinite potential well with confined electrons, each energy level can hold a maximum of two electrons, one with spin up and the other with spin down.
This principle is crucial for correctly calculating ground state energies, as it dictates how electrons fill available energy levels, known as the Aufbau principle.
Ground State Energy
In quantum mechanics, the ground state energy is the lowest possible energy that a quantum mechanical system can have. For a system of electrons, this involves distributing the electrons among available energy levels while respecting the Pauli Exclusion Principle and spin multiplicity, to ensure all are in the lowest possible energy configuration.
  • Each occupied level contributes to the total ground state energy.
  • Calculating the ground state energy for a multi-electron system in an infinite potential well involves adding up the energies of all occupied states.
  • For our example, the ground state energy was calculated as 58 times the factor \(\frac{h^2}{8mL^2}\).
These calculations help predict the physical behavior of electrons in confined systems, like quantum dots or electrons in a metal.
Electron Spin
Electron spin is a fundamental property of electrons, similar in concept to angular momentum in classical mechanics. However, in quantum mechanics, it is intrinsic and quantized. Each electron has a spin of either \(+\frac{1}{2}\) or \(-\frac{1}{2}\), often denoted as 'spin up' or 'spin down.'
  • This property is crucial when considering the filling of energy levels in an infinite potential well.
  • Spins allow for the Pauli Exclusion Principle to be upheld, as two electrons with opposite spins can share the same spatial state.
  • Spin is also responsible for phenomena like magnetism at the quantum level.
Recognizing electron spin's effect on energy states is pivotal for understanding complex systems in quantum mechanics, including solid-state physics and many-body problems.

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Most popular questions from this chapter

An electron is in a state with \(\ell=3\). (a) What multiple of \(\hbar\) gives the magnitude of \(\vec{L} ?\) (b) What multiple of \(\mu_{\mathrm{B}}\) gives the magnitude of \(\vec{\mu} ?(\mathrm{c})\) What is the largest possible value of \(m_{e}\), (d) what multiple of \(\hbar\) gives the corresponding value of \(L_{z}\), and (e) what multiple of \(\mu_{\mathrm{B}}\) gives the corresponding value of \(\mu_{\text {orb } z}\) ? (f) What is the value of the semiclassical angle \(\theta\) between the directions of \(L\), and \(\vec{L}\) ? What is the value of angle \(\theta\) for \((\mathrm{g})\) the second largest possible value of \(m_{\ell}\) and (h) the smallest (that is, most negative) possible value of \(m_{i}\) ?

In 1911, Ernest Rutherford modeled an atom as being a point of positive charge Ze surrounded by a negative charge \(-Z e\) uniformly distributed in a sphere of radius \(R\) centered at the point. At distance \(r\) within the sphere, the electric potential is $$ V=\frac{Z e}{4 \pi \varepsilon_{0}}\left(\frac{1}{r}-\frac{3}{2 R}+\frac{r^{2}}{2 R^{3}}\right) $$ (a) From this formula, determine the magnitude of electric field for \(0 \leq r \leq R .\) What are the (b) electric field and (c) potential for \(r \geq R\) ?

A laser emits at \(424 \mathrm{~nm}\) in a single pulse that lasts \(0.500 \mu \mathrm{s}\). The power of the pulse is \(2.80 \mathrm{MW}\). If we assume that the atoms contributing to the pulse underwent stimulated emission only once during the \(0.500 \mu \mathrm{s}\), how many atoms contributed?

The wavelength of the \(K_{\alpha}\) line from iron is \(193 \mathrm{pm}\). What is the energy difference between the two states of the iron atom that give rise to this transition?

In the subshell \(\ell=3\), (a) what is the greatest (most positive) \(m_{e}\) value, (b) how many states are available with the greatest \(m_{\ell}\) value, and (c) what is the total number of states available in the subshell?
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