Question

State whether each of the following statements is true or false a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced. b) In an infinite one-dimensional potential well, the energy levels are evenly spaced. c) The minimum total energy possible for a classical harmonic oscillator is zero. d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state \((n=0)\) of the one-dimensional quantum harmonic oscillator should also be zero. e) The \(n=0\) state of the one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty \(\Delta x \Delta p\)

Short answer

Answer: True

Step-by-step solution

a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced.

True. The energy levels of a one-dimensional quantum harmonic oscillator follow the formula \(E_n = (n+\frac{1}{2})\hbar\omega\), where \(n = 0, 1, 2, 3,...\) is the quantum number, \(\hbar\) is the reduced Planck's constant, and \(\omega\) is the angular frequency. As we can see, the energy levels differ by a constant value of \(\hbar\omega\) for each step in \(n\). Therefore, the energy levels are evenly spaced.

b) In an infinite one-dimensional potential well, the energy levels are evenly spaced.

False. The energy levels in an infinite one-dimensional potential well follow the formula \(E_n = \frac{n^2 \pi^2\hbar^2}{2mL^2}\), where \(n = 1, 2, 3, ...\) is the quantum number, \(\hbar\) is the reduced Planck's constant, \(m\) is the mass of the particle, and \(L\) is the width of the well. As \(n\) increases, the difference between consecutive energy levels increases. Therefore, the energy levels are not evenly spaced.

c) The minimum total energy possible for a classical harmonic oscillator is zero.

True. For a classical harmonic oscillator, the total energy consists of kinetic energy and potential energy. The minimum total energy occurs when the potential energy is at its minimum and the kinetic energy is zero (the particle is at rest). In this situation, the minimum total energy is zero.

d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state \((n=0)\) of the one-dimensional quantum harmonic oscillator should also be zero.

False. The correspondence principle states that the behavior of a quantum system should approach the classical behavior in the limit of large quantum numbers. This doesn't mean that the minimum possible total energy for the quantum harmonic oscillator should be zero. In fact, the ground state energy of a one-dimensional quantum harmonic oscillator is given by the formula \(E_0 = \frac{1}{2}\hbar\omega\), which is non-zero.

e) The \(n=0\) state of the one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty \(\Delta x \Delta p\)

True. According to the Heisenberg uncertainty principle, the product of the uncertainties in position and momentum has a minimum value \(\Delta x \Delta p \geq \frac{1}{2}\hbar\). For a one-dimensional quantum harmonic oscillator, the ground state (when \(n=0\)) achieves this minimum possible uncertainty. In this state, the wave function represents a Gaussian distribution, which optimally satisfies the uncertainty principle.

Key concepts

Energy Levels

Understanding energy levels in quantum systems is essential for grasping the nature of quantum mechanics. In a one-dimensional quantum harmonic oscillator, the energy levels are indeed evenly spaced. The energy in this system is described by the formula \(E_n = (n+\frac{1}{2})\hbar\omega\), where \(n\) represents the quantum number. Here, \(\hbar\) is the reduced Planck's constant and \(\omega\) stands for the angular frequency of the oscillator. The evenly spaced nature is due to the term \(\hbar\omega\), which adds the same amount of energy for each consecutive level.

However, when dealing with an infinite potential well, the scenario changes. The energy levels follow \(E_n = \frac{n^2 \pi^2\hbar^2}{2mL^2}\). This indicates that the spacing between levels increases as \(n\) grows larger. Hence, the energy levels are not evenly spaced in an infinite potential well. This difference in spacing is due to the \(n^2\) dependence, which causes a quadratic increase in energy with respect to the quantum number.

Infinite Potential Well

The infinite potential well is a fundamental concept in quantum mechanics, illustrating how quantum particles behave under constraints. In this model, a particle is confined to a box with perfectly rigid walls, creating a situation where its wave function is zero outside the well, forcing specific allowed energy levels inside.

The energy levels in an infinite potential well are determined by \(E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}\). This expression shows that as the quantum number \(n\) increases, the energy levels rise in a nonlinear (quadratic) manner, explaining why they aren't evenly spaced. This contrasts with the linear spacing observed in the quantum harmonic oscillator. This characteristic behavior is crucial in understanding the quantization of energy states in constrained quantum systems.

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a cornerstone of quantum mechanics, positing that certain pairs of physical properties, like position and momentum, cannot simultaneously be measured with arbitrary precision. The principle is mathematically represented as \(\Delta x \Delta p \geq \frac{1}{2}\hbar\).

This principle highlights the concept of intrinsic uncertainty in quantum states, emphasizing that the more precisely one property is measured, the less precisely the other can be ascertained. For the ground state \((n=0)\) of a quantum harmonic oscillator, the system reaches this minimal uncertainty relationship. The ground state is often represented by a Gaussian wave function, which is adept at achieving this precise balance of measurement in position and momentum.

Correspondence Principle

The correspondence principle bridges the gap between classical and quantum mechanics. It asserts that the predictions of quantum mechanics should reconcile with classical physics in the limit of large quantum numbers. This principle implies that as systems grow larger or as quantum numbers increase, the quantum behavior appears more classical.

In the context of a quantum harmonic oscillator, while one might expect based on classical intuition that the minimum possible total energy \(E_0\) should be zero, quantum mechanics dictates otherwise. The ground state energy is \(E_0 = \frac{1}{2}\hbar\omega\), which is not zero. This non-zero ground state energy, termed "zero-point energy," underscores how quantum systems retain properties absent in classical systems, even as they transition toward classicality. This highlights the distinctive characteristics of quantum systems, even when they approximate classical behavior.