Question

Determine the expectation value of the position of a harmonic oscillator in its ground state.

Short answer

The expectation value of the position of a harmonic oscillator in its ground state is zero.

Step-by-step solution

Identify the Hamiltonian and the corresponding wavefunction of the ground state

The Hamiltonian for a quantum mechanical harmonic oscillator is given by: $$H=\frac{p^2}{2m}+\frac{1}{2}m{\omega }^2{x}^2$$ where $p$ is the momentum of the particle, $m$ is the mass, $\omega$ is the angular frequency and $x$ is the position. The general solution for the Schrödinger equation with this Hamiltonian yields the wavefunction for the nth energy level. For the ground state (n=0), the normalized wavefunction is: $${\psi }_0(x)=\frac{1}{\sqrt{\pi }\cdot \sqrt[4]{\hslash \cdot m\cdot \omega }}{e}^{-\frac{m\cdot \omega \cdot x^2}{2\cdot \hslash }}$$

Calculating the Expectation Value

The expectation value of the position observable for a quantum state described by the wavefunction $\psi$ is given by $⟨x⟩=\int dxx|\psi {|}^2$. Substituting in our wavefunction for the ground state of the harmonic oscillator, this becomes: $$⟨x⟩=\int dxx|\frac{1}{\sqrt{\pi }\cdot \sqrt[4]{\hslash \cdot m\cdot \omega }}{e}^{-\frac{m\cdot \omega \cdot x^2}{2\cdot \hslash }}{|}^2$$

Solve the Integral

The integral is a Gaussian integral with an odd power of $x$, which gives zero. Hence, the expectation value $$⟨x⟩=0$$ This result is expected because the potential is symmetric about $x=0$ and the ground state is also symmetric about this point. That is, it's just as likely to find the particle to the left of zero as it is to find it to the right in this symmetric state.

Key concepts

Harmonic Oscillator

The harmonic oscillator is a fundamental concept in physics that describes a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. When you think of a spring with a mass attached to it, vibrating back and forth, you're picturing a classic example of a harmonic oscillator.

In quantum mechanics, the quantum harmonic oscillator is particularly important because it provides a solvable model that can be applied to various systems, including molecular vibrations and quantum fields. The energy levels of a quantum harmonic oscillator are quantized, meaning that the system can only possess certain discrete energy values. These energies are given by $E_n=\hslash \omega (n+\frac{1}{2})$ where $n$ is a non-negative integer representing the 'nth' energy level, $\hslash$ is the reduced Planck's constant, and $\omega$ is the angular frequency of the oscillator.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies the behavior of matter and energy at the smallest scales, typically atomic and subatomic levels. It's a theory that extends beyond the limitations of classical physics, providing explanations for phenomena such as the wave-particle duality, quantization of energy, and uncertainty in measurement.

One of the key features of quantum mechanics is the wavefunction, which encodes all the information about a system's state. Interestingly, the wavefunction provides probabilities for the possible outcomes of measurements on the system, a departure from the certainties you'd find in classical physics. The ability to predict the likelihood of finding a particle in a certain position comes down to the complex square of the wavefunction.

Schrödinger Equation

The Schrödinger equation is the fundamental equation of quantum mechanics that describes how the quantum state of a physical system changes over time. In essence, it's a mathematical equation that relates the wavefunction of a system to its energy and time evolution.

For stationary states, such as the ground state of a harmonic oscillator, we use the time-independent Schrödinger equation, which is concerned only with the spatial variables and energy levels. It is pivotal for determining the allowed energy levels of quantum systems and their wavefunctions. For the harmonic oscillator in particular, solving the Schrödinger equation provides us with distinct wavefunctions for each energy level, which are crucial in determining properties like the expectation value of position.

Wavefunction

In quantum mechanics, the wavefunction is at the core of understanding physical systems. It's a mathematical function that describes the quantum state of a particle or system of particles. Notably, the wavefunction is complex-valued and can be used to calculate the probability of finding a particle in a particular region of space upon measurement.

The squared magnitude of the wavefunction, $|\psi {|}^2$, gives the probability density. Moreover, the wavefunctions are typically required to be normalized, ensuring that the total probability across all space is one. In the context of the harmonic oscillator, the ground state wavefunction has a Gaussian form, which reflects the system's symmetry and leads to an expectation value of the position being zero.

Gaussian Integral

The Gaussian integral, also known as the integral of the Gaussian function, is widely used in probability theory, statistics, and many areas of physics, including quantum mechanics. It's expressed as ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-a{x}^2}dx$ and evaluates to $\sqrt{\frac{\pi }{a}}$ for real, positive $a$.

The integral is especially important when dealing with wavefunctions that describe the probability density of particles' positions. The Gaussian nature of certain wavefunctions makes it possible to perform these integrals analytically. As for the expectation value problem, since the position of a harmonic oscillator is multiplied by the probability density—which has symmetrical properties about the central point—integrating an odd function over symmetric limits results in a zero value, reinforcing the concept of symmetry in the system.