Question

Simple harmonic oscillation occurs when the potential energy function is equal to \((1 / 2) k x^{2},\) where \(k\) is a constant. What happens to the ground state energy level if \(k\) is increased? a) It increases. b) It remain the same. c) It decreases.

Short answer

Answer: The ground state energy level increases when \(k\) is increased.

Step-by-step solution

Recall the formula for the ground state energy level of a quantum harmonic oscillator

For a quantum mechanical harmonic oscillator, the energy level can be written as: \[E_{n}=(n+\frac{1}{2})\hbar\omega\] where \(n\) is a non-negative integer, \(\hbar\) is the reduced Planck's constant, and \(\omega\) is the angular frequency. The ground state energy level, denoted as \(E_0\), occurs when \(n=0\). Therefore, we can write the ground state energy level as: \[E_{0}=\frac{1}{2}\hbar\omega\]

Express the angular frequency in terms of the spring constant \(k\)

The angular frequency \(\omega\) of the simple harmonic oscillator is related to the spring constant \(k\) and the mass \(m\) of the oscillator as follows: \[\omega = \sqrt{\frac{k}{m}}\]

Substitute the angular frequency expression in the ground state energy level formula

Now we will substitute the expression for \(\omega\) from step 2 into the ground state energy level formula from step 1: \[E_0=\frac{1}{2}\hbar\sqrt{\frac{k}{m}}\]

Analyze the dependence of the ground state energy level on the spring constant \(k\)

From the expression derived in step 3, we can see that \(E_0\) is proportional to the square root of the spring constant \(k\). More specifically, if the spring constant \(k\) is increased, the ground state energy level \(E_0\) will also increase due to their direct proportionality. Thus, the correct answer to the question is: a) The ground state energy level increases when \(k\) is increased.

Key concepts

Simple Harmonic Oscillation

An intuitive way to understand simple harmonic oscillation is to envision a ball attached to a spring. When the ball is pulled and released, it bobs back and forth about its equilibrium position. This type of motion is characterized by its repetitive nature and is described mathematically by the force being proportional to the displacement but in the opposite direction. In mathematical terms, Hooke's Law is expressed as:
\[ F = -k x \]
where \( F \) is the restoring force exerted by the spring, \( k \) is the spring constant that measures the stiffness of the spring, and \( x \) is the displacement of the ball from its equilibrium position.

It's important to note that in this scenario, when the force is translated into potential energy, the function becomes:
\[ U(x) = \frac{1}{2} k x^2 \]
This formula represents the potential energy at a given displacement in simple harmonic motion. The energy is solely dependent on the position of the object and the spring constant, with the energy being highest at the maximum displacement. The beauty of simple harmonic motion lies in its predictability and the way it can be easily charted as a sine or cosine function over time, demonstrating periodic motion that is fundamental to understanding various physical systems, from playground swings to molecular vibrations.

Quantum Harmonic Oscillator

Delving into the quantum realm, the quantum harmonic oscillator serves as a cornerstone model in quantum mechanics. At the quantum level, particles such as electrons don't behave like a classical ball and spring; instead, they have wave-like properties, and their energies are quantized. This means that unlike a classical system where the ball can have any arbitrary amount of energy, a quantum oscillator has distinct, well-defined energy levels.

These energy levels are determined by the formula:
\[ E_n = \bigg(n + \frac{1}{2}\bigg)\bar{h}\frac{\text{The ground state energy level}}{2} \]
Here, \( E_n \) is the energy of the nth level, \( \bar{h} \) is the reduced Planck's constant, and \( \text{The ground state energy level} \) is the frequency of the oscillation. The ground state energy level, represented by \( n = 0 \), is analogous to the lowest step on a ladder that the particle can occupy.

One peculiar aspect of the quantum harmonic oscillator is that it possesses a non-zero ground state energy, known as zero-point energy. This means that even in the lowest energy state, the system still has some intrinsic motion—a fascinating deviation from classical expectations where motion ceases at zero energy.

Spring Constant

The spring constant, symbolized by \( k \), is a measure of the stiffness of a spring. It dictates how much force is needed to stretch or compress the spring by a certain amount. A higher spring constant indicates a stiffer spring that requires more force to be deformed. Conversely, a lower spring constant means the spring is softer and easier to stretch or compress.

The relationship between the spring constant and the force applied to the spring is governed by Hooke's Law, as mentioned earlier. A critical aspect of this is that the spring exerts a force that is equal in magnitude but opposite in direction to the applied force, embodying an example of Newton's third law of motion. If the spring's constant increases, it means that for the same displacement, a greater force is required to hold the spring elongated or compressed. This implies a stiffer spring, which in turn affects the dynamics of systems that involve springs, such as the timing mechanism in watches or the handling of a vehicle's suspension system.

Understanding the spring constant is fundamental when studying oscillatory systems, especially in the quantum perspective where the energy levels are influenced by the stiffness of the potential well, akin to the stiffness of a spring, as shown in the ground state energy of a quantum harmonic oscillator.