Question

Example 37.1 calculates the energy of the wave function with the lowest quantum number for an electron confined to a box of width \(2.00 \AA\) in the one-dimensional case. However, atoms are three-dimensional entities with a typical diameter of \(1.00 \AA=10^{-10} \mathrm{~m} .\) It would seem then that the next, better approximation would be that of an electron trapped in a three-dimensional infinite potential well (a potential cube with sides of \(1.00 \mathrm{~A}\) ). a) Derive an expression for the electron wave function and the corresponding energies for a particle in a three dimensional rectangular infinite potential well. b) Calculate the lowest energy allowed for the electron in this case.

Short answer

The electron's wave function trapped in a three-dimensional infinite potential well with side lengths of 1.00 Å is given by: \(\Psi(x, y, z) = \sqrt{\frac{8}{L_x L_y L_z}}\sin(\frac{n_x\pi x}{L_x})\sin(\frac{n_y\pi y}{L_y})\sin(\frac{n_z\pi z}{L_z})\) where \(n_x\), \(n_y\), and \(n_z\) are positive integers, and \(L_x = L_y = L_z = 10^{-10}\mathrm{~m}\). The lowest energy allowed for the electron in the three-dimensional infinite potential well is approximately \(1.76 \times 10^{-18}\) Joules.

Step-by-step solution

Setup Schrödinger's Equation for the 3D infinite potential well

Consider an electron trapped in an infinite potential well of dimensions \(L_x = L_y = L_z = 1.00 \mathrm{~A} = 10^{-10}\mathrm{~m}\). For the particle to be confined, the potential energy must be: - V(x, y, z) = 0 for 0 < x < \(L_x\), 0 < y < \(L_y\), 0 < z < \(L_z\) - V(x, y, z) = ∞ for any other x, y, or z values. The Schrödinger equation that governs the particle's wave function is given by: \(-\frac{\hbar^2}{2m}(\frac{\partial^2\Psi}{\partial x^2}+\frac{\partial^2\Psi}{\partial y^2}+\frac{\partial^2\Psi}{\partial z^2})+V(x,y,z)\Psi = E\Psi\) For our infinite potential well, we can rewrite this as: \(-\frac{\hbar^2}{2m}(\frac{\partial^2\Psi}{\partial x^2}+\frac{\partial^2\Psi}{\partial y^2}+\frac{\partial^2\Psi}{\partial z^2}) = E\Psi\) **Step 2: Solve the Schrödinger equation using separation of variables**

Solve the 3D Schrödinger equation

Using the method of separation of variables, we assume that the wave function can be written as a product of three single-variable functions: \(\Psi(x, y, z) = X(x)Y(y)Z(z)\) Substituting this into the equation above, we get: \(-\frac{\hbar^2}{2m}(\frac{1}{X}\frac{\partial^2 X}{\partial x^2} + \frac{1}{Y}\frac{\partial^2 Y}{\partial y^2} + \frac{1}{Z}\frac{\partial^2 Z}{\partial z^2}) = E\Psi\) Now, divide both sides by \(\Psi\), then separate it into three different equations, each of them representing components x, y, and z: \(\frac{-\frac{\hbar^2}{2m}(\frac{1}{X}\frac{\partial^2 X}{\partial x^2})}{X(x)Y(y)Z(z)} + \frac{-\frac{\hbar^2}{2m}(\frac{1}{Y}\frac{\partial^2 Y}{\partial y^2})}{X(x)Y(y)Z(z)} + \frac{-\frac{\hbar^2}{2m}(\frac{1}{Z}\frac{\partial^2 Z}{\partial z^2})}{X(x)Y(y)Z(z)} =\frac{E}{X(x)Y(y)Z(z)}\) Therefore, we have: \(-\frac{\hbar^2}{2m}\frac{1}{X}\frac{\partial^2 X}{\partial x^2} = E_x X\) \(-\frac{\hbar^2}{2m}\frac{1}{Y}\frac{\partial^2 Y}{\partial y^2} = E_y Y\) \(-\frac{\hbar^2}{2m}\frac{1}{Z}\frac{\partial^2 Z}{\partial z^2} = E_z Z\) Solving each equation, we find: \(X(x) = \sqrt{\frac{2}{L_x}}\sin(\frac{n_x\pi x}{L_x})\) \(Y(y) = \sqrt{\frac{2}{L_y}}\sin(\frac{n_y\pi y}{L_y})\) \(Z(z) = \sqrt{\frac{2}{L_z}}\sin(\frac{n_z\pi z}{L_z})\) Where \(n_x, n_y, n_z\) are positive integers (\(1, 2, 3,\)...) and the corresponding energies \(E_x, E_y\), and \(E_z\) are given by: \(E_x = \frac{n_x^2\pi^2\hbar^2}{2mL_x^2}\) \(E_y = \frac{n_y^2\pi^2\hbar^2}{2mL_y^2}\) \(E_z = \frac{n_z^2\pi^2\hbar^2}{2mL_z^2}\) Finally, finding the particle's wave function \(\Psi(x, y, z)\) and total energy \(E\):

Determine the wave function and energy expression

The three-dimensional wave function: \(\Psi(x, y, z) = X(x)Y(y)Z(z) = \sqrt{\frac{8}{L_x L_y L_z}}\sin(\frac{n_x\pi x}{L_x})\sin(\frac{n_y\pi y}{L_y})\sin(\frac{n_z\pi z}{L_z})\) The total energy is the sum of \(E_x, E_y\), and \(E_z\): \(E = E_x + E_y + E_z = \frac{\pi^2\hbar^2}{2m}( \frac{n_x^2}{L_x^2}+ \frac{n_y^2}{L_y^2}+ \frac{n_z^2}{L_z^2})\) Now that we have the wave function and energy expressions, we can proceed with finding the lowest energy allowed for the electron. **Step 3: Find the lowest energy allowed**

Determine the lowest energy state

The lowest energy state corresponds to the lowest possible values of \(n_x, n_y, n_z\), which are equal to 1. Therefore, let: \(n_x = n_y = n_z = 1\) Substitute the values of \(n_x\), \(n_y\) and \(n_z\) into the energy expression: \(E = \frac{\pi^2\hbar^2}{2m}( \frac{1}{L_x^2}+ \frac{1}{L_y^2}+ \frac{1}{L_z^2})\) Now, we can use the values of \(L_x = L_y = L_z = 10^{-10}\mathrm{~m}\), \(\hbar = 1.05457182 × 10^{-34} J⋅s\) (reduced Planck constant), and \(m = 9.10938356 × 10^{-31} kg\) (electron mass):

Calculate the lowest energy allowed

Insert the values to compute the lowest energy allowed: \(E = \frac{\pi^2(1.05457182 × 10^{-34})^2}{2(9.10938356 × 10^{-31})(10^{-10})^2} (3)\) Solving for E, we get: \(E \approx 1.76 \times 10^{-18} J\) Thus, the lowest energy allowed for the electron in this three-dimensional infinite potential well is approximately \(1.76 \times 10^{-18}\) Joules.

Key concepts

Three-Dimensional Infinite Potential Well

When thinking about quantum mechanics, it helps to start with the concept of an infinite potential well. This is an imaginary box with impenetrable walls. Particles, such as electrons, are trapped inside and can't escape. The walls have infinitely high potential energy, which means the particle is "locked" in the box where the potential energy is zero. Now, consider this box in three dimensions, so that the particle can move along the x, y, and z directions.

Lengths of the sides of this three-dimensional box are often denoted as \( L_x \), \( L_y \), and \( L_z \). In our problem, these lengths are all \( 1.00 \, Å = 10^{-10} \, \text{m} \). Movement is confined to inside the box, where the potential energy \( V(x, y, z) \) is zero. Outside, it jumps to infinity.

For a particle inside this well, the allowed energies and associated wave functions are determined by solving the Schrödinger equation, as we'll delve into next.

Schrödinger equation

The Schrödinger equation is the heart of quantum mechanics, describing how quantum states evolve over time. In our infinite potential well scenario, we use a time-independent version to find the wave function of the particle. It's a differential equation and a powerful tool for understanding quantum systems. For three dimensions, it is expressed as: \[ -\frac{\hbar^2}{2m} \left(\frac{\partial^2\Psi}{\partial x^2} + \frac{\partial^2\Psi}{\partial y^2} + \frac{\partial^2\Psi}{\partial z^2}\right) + V(x, y, z)\Psi = E\Psi \] Where:

• \( \hbar \) is the reduced Planck's constant.
• \( m \) is the mass of the electron.
• \( E \) is the energy of the electron within the well.

For a three-dimensional infinite well, \( V(x, y, z) = 0 \) inside the box and is zero. Outside, it's limitless. The solution to this equation gives us both the energy levels of the electron and its wave function, an equation we'll look into more deeply shortly. The method of separation of variables helps us find solutions by splitting the equation into manageable pieces, each depending only on one variable.

Quantum Energy Levels

The concept of quantum energy levels arises directly from solving the Schrödinger equation for our infinite potential well. These energy levels represent discrete amounts of energy that a particle like an electron can have in a quantum system, as opposed to the continuous energy values in classical physics.

In a three-dimensional infinite potential well, these energy levels depend on three quantum numbers \( n_x, n_y, n_z \), representing each spatial dimension. These numbers must be positive integers, such as 1, 2, 3, and so on. The energy for each state is given by: \[ E = \frac{\pi^2\hbar^2}{2m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) \] Each combination of \( n_x \), \( n_y \), and \( n_z \) gives a unique energy level. The lowest quantum energy level is generally when \( n_x = n_y = n_z = 1 \). This represents the ground state of the system, the lowest possible energy state the system can occupy. That is the focus of our calculation for energy.

Electron Wave Function

The electron wave function \( \Psi(x, y, z) \) provides a description of the electron's quantum state in the potential well. It is derived as part of solving the Schrödinger equation for the system. This function is crucial because its square (\( |\Psi|^2 \)) gives the probability density of finding the electron at a specific location within the well.

The wave function for an electron in a three-dimensional infinite potential well is represented as a product of sine functions: \[ \Psi(x, y, z) = \sqrt{\frac{8}{L_x L_y L_z}} \sin\left(\frac{n_x\pi x}{L_x}\right) \sin\left(\frac{n_y\pi y}{L_y}\right) \sin\left(\frac{n_z\pi z}{L_z}\right) \]

• The \( \sin \) terms capture the standing wave patterns inside the box.
• The quantum numbers \( n_x, n_y, n_z \) dictate the number of nodes (zero crossings).

The wave function thus not only determines the energy of the electron but also its spatial distribution, reflecting the particle's wave nature characteristic of quantum mechanics.