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Chapter 13: Problem 16

Calculate the energy spectrum for a free electron in a box of sides \(a, a . L\). with \(a

Short Answer

Expert verified
The energy spectrum for a free electron in a box of sides \(a, a, L\) with \(a < L\) is given by: \[ E_{n_x, n_y, n_z} = \frac{h^2}{8m} \left( \frac{n_x^2}{a^2} + \frac{n_y^2}{a^2} + \frac{n_z^2}{L^2} \right) \] For a box with dimensions \(a=1 \, nm\) and \(L=1 \, \mu m\), the spacings between consecutive energy levels \(S_x\), \(S_y\), and \(S_z\) are given by: \[ S_x = S_y = \frac{h^2}{8 m a^2}(2n_x + 1) = \frac{h^2}{8 m a^2}(2n_y + 1) \] \[ S_z = \frac{h^2}{8 m L^2}(2n_z + 1) \] These spacings show that energy levels get closer together as they get higher, emphasizing the quantization of the energy spectrum for a free electron confined in a quantum box.

Step by step solution

01

Calculate the Energy Levels for Given Quantum Numbers

We will start by plugging in the given dimensions of the quantum box into the equation for energy levels: \[ E_{n_x,n_y,n_z} =\frac{h^2}{8m} \left( \frac{n_x^2+d n_y^2}{a^2}+\frac{n_z^2}{L^2}\right) \] where \(a = 1 \, nm, L = 1 \, \mu m, m = 9.11 \times 10^{-31} kg\), and \(h = 6.626 \times 10^{-34} Js\), using units of meters and seconds for dimensions and time respectively.
02

Discuss the Spacings for Given Box Dimensions

The spacing or difference in energy between two consecutive levels in any particular direction is given by the difference \(E_{i+1}-E_i\), where \(i\) represent the quantum numbers \(n_x, n_y, n_z\). By subtracting the formulas for consecutive energy levels and simplifying, expressions for the spacings \(S_x = E_{n_x+1,n_y,n_z} - E_{n_x,n_y,n_z}\), \(S_y = E_{n_x,n_y+1,n_z} - E_{n_x,n_y,n_z}\), and \(S_z = E_{n_x,n_y,n_z+1} - E_{n_x,n_y,n_z}\) can be obtained as: \[ S_x = \frac{h^2(n_x+1)^2}{8 m a^2} - \frac{h^2n_x^2}{8 m a^2} = \frac{h^2}{8 m a^2}(2n_x + 1) \] \[ S_y = \frac{h^2(n_y+1)^2}{8 m a^2} - \frac{h^2n_y^2}{8 m a^2} = \frac{h^2}{8 m a^2}(2n_y + 1) \] \[ S_z = \frac{h^2(n_z+1)^2}{8 m L^2} - \frac{h^2n_z^2}{8 m L^2} = \frac{h^2}{8 m L^2}(2n_z + 1) \] These formulas display how the spacing between energy levels depends on each of the quantum numbers, and also show that the spacings decrease as the quantum number increases. This means that energy levels get closer together as they get higher, which will be an important point in the discussion. With the given dimensions, we can also calculate numerical values for the spacings between the first and second levels in each direction (i.e., for \(n_x = n_y = n_z = 1\)).

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Key Concepts

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

Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces a significant departure from classical mechanics and deals with phenomena that cannot be explained by classical physical laws, such as the behavior of electrons in atoms.

Key to quantum mechanics is the idea that energy is quantized; it can exist only in discrete amounts, or 'quanta'. So, unlike a ball rolling smoothly up and down a slope, tiny subatomic particles like electrons can only exist at certain, specific energy levels. The rules governing these quantized energies are determined by the wave-like nature of particles, leading to the concept of wave functions and the probabilistic interpretation of phenomena.
Particle in a Box
The 'particle in a box' is a model from quantum mechanics, which encapsulates the particle's behavior when it is constrained within a small region of space, such as an electron trapped within a crystal lattice or a molecule. This model assumes a perfectly rigid and immovable boundary and that the potential energy inside the box is zero.

In this scenario, the free particle cannot have any energy value as it would in classic physics; rather, it is limited to certain discrete energy levels. The wave function describing the particle must be zero at the boundaries, leading to specific solutions that are sine and cosine functions standing wave patterns. The conditions that only these specific wave patterns are possible lead to the quantization of energy levels for the particle.
Quantum Numbers
Quantum numbers arise naturally from the mathematics used to describe the energy levels and the quantum state of a particle in quantum mechanics. Each quantum state of a particle is defined by a set of quantum numbers, and these numbers provide important information about the behavior of the particle.

There are several types of quantum numbers, including the principal quantum number (which determines the energy level of an electron in an atom), the orbital angular momentum quantum number, the magnetic quantum number, and the spin quantum number. Each quantum number can take on only certain discrete values. When dealing with a particle in a box, the quantum numbers are positive integers () that label the energy levels and standing wave patterns within the box.
Energy Level Spacings
Energy level spacings refer to the differences in energy between successive quantum states. In the context of a particle in a box, these spacings give insight into how tightly packed the energy levels are. The spacings are especially important for understanding the physical properties of systems at the quantum level, like the absorption and emission spectra of atoms.

Typically, the energy level spacings decrease as the energy increases because the higher the energy level, the less the difference becomes between it and the next level. This is reflected in the formulas for the spacings between energy levels (). The calculated values for the spacings demonstrate the quantum mechanical principle that energy levels are not evenly spaced and reveal the influence of the dimensions of the confinement on the energy of the particle.

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Most popular questions from this chapter

Consider two electrons in the same spin state, interacting with a potential $$ \begin{aligned} v\left(x_{1}-x_{2} \mid\right) &=-V_{0} & &\left|x_{1}-x_{2}\right| \leq a \\ &=0 & & \text { elsewhere } \end{aligned} $$ What is the lowest energy of the two-electron state, assuming that the total momentum of the two electrons is zero? Assume that the potential is deep enough for more than one bound state. (Hint: Reduce the problem to a one- particle equation with reduced mass. Remember that when the elec. trons are in the same spin state, the spatial wave function must change sign when \(x_{1} \leftrightarrow x_{2}\). \()\)

Consider two electrons described by the Hamiltonian $$ \left.H=\frac{p_{1}^{\prime}}{2 m}+\frac{p_{3}^{2}}{2 m}+V x_{0}\right)+V\left(x_{2}\right) $$ where \(V(x)=\infty\) for \(x<0\) and for \(x>a, V(x)=0\) for \(0

A nucleus consists of \(N\) neutrons and \(Z\) protons, with \(N+Z=A\). If the radius of the nucleus is given by \(R=r_{0} A^{i / 3}\) with \(r_{0}=1.1 \mathrm{fm}\left(1 \mathrm{fm}=10^{-13} \mathrm{~m}\right)\) and if the neutron and proton masses are treated as equal, \(\left(1.7 \times 10^{-27} \mathrm{~kg}\right)\), write an expanssion for the Fermi energy of the proton "gas" land the neutron "gas, "assuming that the protons and neutrons move as free particles. What are the Fermi energies if \(N=126\) and \(Z=82 ?\)

Calculate the degencracy of states in a cubic box of volume \(L^{3}\) as a function of \(E\); that is, calculate the number of states in the interval \((E, E+d E)\), and use this to obtain the density of states of as electron gas, keeping in mind that there are two electrons per energy state. (Hint: How many \(\left\\{n_{1}, n_{y}\right.\), \(n_{\mathrm{l}} \mid\) are there for which \(\Sigma n_{3}^{2}=2 m E L^{2} h^{2} n^{2} 7\) )

Prove that the exchange operator \(P_{n}\) is hermitian.
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