Question

An electron is confined in a three-dimensional cubic space of \(L^{3}\) with infinite potentials. a) Write down the normalized solution of the wave function for the ground state. b) How many energy states are available between the ground state and the second excited state? (Take the electron's spin into account.)

Short answer

Answer: The normalized wave function for the ground state of an electron in a 3D cubic box is given by \(\Psi_{1,1,1}(x,y,z) = \left(\frac{2}{L}\right)^{3/2} \sin\left(\frac{\pi x}{L}\right)\sin\left(\frac{\pi y}{L}\right)\sin\left(\frac{\pi z}{L}\right)\). There are 6 energy states available between the ground state and the second excited state when considering the electron's spin.

Step-by-step solution

Find the general solution of the wave function

Given an electron in a three-dimensional cubic space of \(L^{3}\) with infinite potentials, the wave function for every state is given by: \[ \Psi_{n_x,n_y,n_z}(x,y,z) = \left(\frac{2}{L}\right)^{3/2} \sin\left(\frac{n_x \pi x}{L}\right)\sin\left(\frac{n_y \pi y}{L}\right)\sin\left(\frac{n_z \pi z}{L}\right) \] where \(n_x\), \(n_y\), and \(n_z\) are the quantum numbers for the dimensions x, y, and z, respectively.

Write down the normalized wave function for the ground state

The ground state is the state with the smallest quantum numbers (\(n_x=1, n_y=1, n_z=1\)). So, the normalized wave function for the ground state is: \[ \Psi_{1,1,1}(x,y,z) = \left(\frac{2}{L}\right)^{3/2} \sin\left(\frac{\pi x}{L}\right)\sin\left(\frac{\pi y}{L}\right)\sin\left(\frac{\pi z}{L}\right) \] This is the answer for part (a).

Calculate the energy for each quantum state

The energy of a particle in a 3D box in terms of its quantum numbers is given by: \[ E_{n_x,n_y,n_z} = \frac{h^2}{8mL^2} (n_x^2 + n_y^2 + n_z^2) \] where \(h\) is Planck's constant and \(m\) is the mass of the electron.

Find the energy states between the ground state and the second excited state

From the energy equation, we have the energies of the ground and second excited states: - Ground state: \(E_{1,1,1}\) with quantum numbers \((1,1,1)\) - Second excited state: \(E_{2,0,0}\) with quantum numbers \((2,0,0)\) Now, we must count the number of energy states between these two levels. By checking the possible quantum number combinations and their corresponding energies, we find that the only available energy states between the ground state and the second excited state are: 1. \(E_{1,1,2}\) with quantum numbers \((1,1,2)\) 2. \(E_{1,2,1}\) with quantum numbers \((1,2,1)\) 3. \(E_{2,1,1}\) with quantum numbers \((2,1,1)\) Since we also need to take the electron's spin into account, each of the three available energy states can accommodate 2 electrons with opposite spins. Thus, there are a total of 6 energy states between the ground state and the second excited state. The answer for part (b) is 6 energy states.

Key concepts

Wave Function Normalization

Wave function normalization is an essential concept in quantum mechanics which ensures that the total probability of finding a particle within its entire space is always one. This normalization condition arises because the probability of a particle's presence at a location is represented by the square of the magnitude of its wave function. For a three-dimensional cubic space, the normalization factor is typically a multiple of \(\frac{2}{L}\)^{3/2}, where \(L\) is the length of the cube's side.

During normalization, we adjust the wave function so that the integral of the squared wave function over all space is equal to one. For the electron confined in a 3D cubic box, the normalized wave function for the ground state can be expressed as \(\Psi_{1,1,1}(x,y,z) = \left(\frac{2}{L}\right)^{3/2} \sin\left(\frac{\pi x}{L}\right)\sin\left(\frac{\pi y}{L}\right)\sin\left(\frac{\pi z}{L}\right)\). With this adjusted wave function, we ensure that the probability of finding the electron anywhere in the cube sums up to unity, which aligns with the fundamental postulates of quantum mechanics.

Quantum Numbers

Quantum numbers are vital when discussing quantum systems as they define the distinct energy states of quantum particles like electrons. In a three-dimensional space, especially for particles in a box, there are three quantum numbers corresponding to each spatial dimension: \(n_x\), \(n_y\), and \(n_z\).

These quantum numbers describe the quantization of energy levels and are always integer values. They essentially count the number of wave function nodes along each direction inside the box, indicating different states of motion for the particle and accordingly their energy. The ground state, where the electron has the lowest energy, corresponds to quantum numbers \(n_x=1, n_y=1, n_z=1\), which is the standard starting point for most problems involving particles in a box.

Particle in a 3D Box

When dealing with a particle in a 3D box—a cornerstone scenario in quantum mechanics—we assume the particle is free to move within a cubic space while being completely contained by infinitely high potential barriers. This model simplifies the three-dimensional motion of the particle and is described using a wave function depending on three variables \(x\), \(y\), and \(z\), each associated with a specific quantum number.

The boundaries of the box impose strict conditions on the wave function, causing it to nullify at the edges and leading to a discrete set of allowed energies, unlike the continuous spectrum in classical mechanics. This idealized system dramatically illustrates one of quantum mechanics' most peculiar features: quantization of energy levels.

Electron Energy States

In quantum mechanics, electron energy states in a system like a 3D box are determined by the quantum numbers of the electron. Each set of quantum numbers corresponds to a unique energy level given by the formula \(E_{n_x,n_y,n_z} = \frac{h^2}{8mL^2} (n_x^2 + n_y^2 + n_z^2)\), where \(h\) represents Planck's constant and \(m\) the mass of the electron.

Between the ground state and any excited state, multiple energy levels can be found, depending on the quantum numbers. Importantly, due to the electron's spin, each energy level can accommodate two electrons with opposite spins. This point is crucial when considering the number of energy states available between given levels, like the ground state and the second excited state. For example, there are six such energy states between the ground state and the second excited state in a cubic box, since each of the three spatially distinct levels can hold two electrons with opposite spins.