Question

The ground state energy of a particle of mass \(m\) with potential energy \(U(x)=U_{0} \cosh (x / a),\) where \(U_{0}\) and \(a\) are constants. Show that the ground state energy of the particle can be estimated as: $$ E_{0} \cong U_{0}+\frac{1}{2} \hbar\left(\frac{U_{0}}{m a^{2}}\right)^{\frac{1}{2}}. $$

Short answer

Question: In a given potential energy of the form \(U(x) = U_0\cosh(x/a)\), with \(U_0\) and \(a\) being constants, show that the ground state energy of a particle can be approximated as \(E_0 \cong U_0 + \frac{1}{2}\hbar\left(\frac{U_0}{ma^2}\right)^{\frac{1}{2}}\), with \(m\) representing the particle's mass. Answer: To approximate the ground state energy, we first write down the time-independent Schrödinger equation for the particle and find the minimum of the potential energy function, \(U(x)\). Next, we expand the potential energy \(U(x)\) by using the Taylor series around the minimum value and substitute it into the Schrödinger equation. The resulting Schrödinger equation resembles the harmonic oscillator equation, which allows us to calculate the ground state energy using the quantum harmonic oscillator energy levels formula. Finally, we find that \(E_0 \cong U_0 + \frac{1}{2}\hbar\left(\frac{U_0}{ma^2}\right)^{\frac{1}{2}}\).

Step-by-step solution

Write down the Time Independent Schrödinger equation for the particle

The time-independent Schrödinger equation for a particle with potential energy \(U(x)\) is given by: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + U(x)\psi(x) = E\psi(x), $$ where \(\hbar\) is the reduced Planck constant, \(\psi(x)\) is the wavefunction of the particle, and \(E\) is the energy eigenvalue.

Find the minimum of \(U(x)\)

The minimum value of the potential energy \(U(x)\) occurs when the derivative of the potential energy \(U'(x)\) is equal to zero. So we find the derivative, and then solve for \(x\): $$ U'(x)=\frac{d}{dx}\left(U_0\cosh\left(\frac{x}{a}\right)\right)=\frac{U_0}{a}\sinh\left(\frac{x}{a}\right). $$ The minimum is found when \(U'(x)=0\), which implies that \(\sinh(\frac{x}{a})=0\). This condition is met when \(x=0\).

Expand \(U(x)\) around its minimum using Taylor series

We can approximate the potential energy \(U(x)\) around the minimum value by expanding it using the Taylor series and keeping only the first-order term, which results in a potential energy with a quadratic term: $$ U(x) \approx U(0) +\frac{1}{2}U''(0)x^2 = U_0+\frac{1}{2}\frac{U_0}{a^2}x^2, $$ where \(U''(0)\) is the second derivative of \(U(x)\) with respect to \(x\) evaluated at the minimum of the potential.

Write the Schrödinger equation with the approximate potential energy

Now we substitute the approximate potential energy \(U(x)\) into the Schrödinger equation, and the equation becomes: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + \left(U_0+\frac{1}{2}\frac{U_0}{a^2}x^2\right)\psi(x) = E\psi(x). $$

Observe that the Schrödinger equation is now a harmonic oscillator equation

With the approximate potential energy, our Schrödinger equation now resembles the equation for a harmonic oscillator with the following parameters: $$ k=\frac{U_0}{a^2},\quad\text{and}\quad \omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{\frac{U_0}{a^2}}{m}}=\sqrt{\frac{U_0}{ma^2}}. $$ The energy levels of a quantum harmonic oscillator are given by $$ E_n = \hbar \omega\left(n+\frac{1}{2}\right), $$ where \(n\) is the energy level index.

Estimate the ground state energy

The ground state energy (\(E_0\)) corresponds to \(n=0\). So, we calculate \(E_0\) using the parameters we found above: $$ E_0 \cong \hbar\omega\left(\frac{1}{2}\right) = \hbar\sqrt{\frac{U_0}{ma^2}}\left(\frac{1}{2}\right) = U_0+\frac{1}{2}\hbar\left(\frac{U_0}{ma^2}\right)^{\frac{1}{2}}. $$ Thus, we have shown that the ground state energy of the particle can be estimated as: $$ E_0 \cong U_0 + \frac{1}{2}\hbar\left(\frac{U_0}{ma^2}\right)^{\frac{1}{2}}. $$

Key concepts

Time Independent Schrödinger Equation

The Time Independent Schrödinger Equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system in a stationary state. It is expressed as:

\[\begin{equation}-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + U(x)\psi(x) = E\psi(x),\end{equation}\]
where \(\hbar\) is the reduced Planck's constant, \(m\) is the mass of the particle, \(\psi(x)\) is the wave function, \(U(x)\) is the potential energy, and \(E\) represents the energy of the system. The equation is time-independent because it does not involve the time variable, focusing on energy levels and stationary states. By solving this equation, you can find the wave functions, which represent the probable positions of particles, and the corresponding energy levels that the quantum system can occupy.

Quantum Harmonic Oscillator

The quantum harmonic oscillator is a quantum analog of the classical harmonic oscillator. It is a model that describes a particle subject to a restoring force proportional to the displacement from an equilibrium position. The potential energy of a classical harmonic oscillator follows Hooke's law and is given by a quadratic function of position:

\[\begin{equation}U(x) = \frac{1}{2}kx^2,\end{equation}\]
with \(k\) being the force constant. In quantum mechanics, the harmonic oscillator model is important as it can be solved exactly using the Schrödinger equation. Its energy levels are quantized and given by:

\[\begin{equation}E_n = \hbar \omega\left(n+\frac{1}{2}\right),\end{equation}\]
where \(\omega\) is the angular frequency of the oscillator, and \(n\) is a non-negative integer representing the energy level. This model applies to various physical systems, including vibrating atoms in a molecule and quantum fields in quantum field theory.

Taylor Series Expansion

Taylor series is a mathematical tool used to approximate functions around a given point, usually near zero. It is central to calculus and numerical analysis, allowing us to write a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In quantum mechanics, the Taylor series is often used to simplify potential functions around their minimum energy, the equilibrium position. Assuming the function is well-behaved and differentiable, the Taylor series around \(x = 0\) is given by:

\[\begin{equation}f(x) \approx f(0) + f'(0)x + \frac{1}{2}f''(0)x^2 + \dots,\end{equation}\]
This expansion is particularly useful in estimating ground state energies in quantum systems by approximating complex potentials with a simpler, quadratic form that resembles the potential of a harmonic oscillator, which is exactly solvable in quantum mechanics.

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 conceptually challenging ideas, such as wave-particle duality, superposition, and quantization of energy. Unlike classical mechanics, where physical quantities are continuous, quantum mechanics describes a discrete and probabilistic nature of matter and energy. One of the core tenets is the wave function, which provides the probabilities of finding a particle in various places and states. The behavior of these particles is governed by the Schrödinger equation, with solutions providing insights into the discrete energy levels and the probability distributions of particles. Quantum mechanics forms the foundation of modern physics and chemistry and has profound implications in technology, including semiconductors and quantum computing.