Question

In Section \(5.5\). it was shown that the infinite well coergis follow simply from \(\Lambda=h / p\), the fornulafor kinelic encrgy. \(p^{2} / 2 m\); and a famous standing-wave condition \(\lambda=2 U_{n} .\) The arguments are perfectly valid when the potential energy is 0 (inside the well) and \(L\) is strictly conskant, but they can also be useful in other cases. The length \(L\) allowed the wave should be roughly the distance between the classical tuming points, where there is no kinetic eneigy left. Apply these arguments to the oscillator potential energy, \(U(x)=\frac{1}{2} \kappa x^{2}\). Find the location \(x\) of the classical tuming point in terms of \(E\); use nwice this distance for L: then insert this into the infinite well eneigy fonnula so that \(E\) appears on both sides. Thus far, the procedure Ieally only deals with kinctic energy. Assume, as is tue for a classical oscillator, that there is as much potential eneigy, on average, as kinetic energy. What do you obtain for the quantized energies?

Short answer

By following these steps, we should be able to find an equation for the energy of the oscillator, with the energy \( E \) expressing energy quantization for specific states \(n\). This shows how quantum and classical physics concepts are intertwined, and how energy quantization comes into effect in an oscillator system.

Step-by-step solution

Determine the Turning Points

The turning points in a classical oscillator is defined as the points where the kinetic energy becomes zero, and the particle motion reverses direction. Here, all the total energy E becomes potential energy. The potential energy of the oscillator is given by \(U(x)=\frac{1}{2} \kappa x^{2}\). Setting this equal to \( E \), we get \(\frac{1}{2} \kappa x^{2} = E\). Solving this equation for x results in a turning point at \( x = \pm \sqrt{2E/ \kappa} \).

Determine the Length of the Well

The turning points above define the end points of motion. So, twice the turning point distance should be the length of the well. Therefore, \(L = 2x = 2 \sqrt{2E/ \kappa } \).

Apply the Infinite Well Energy Formula

Now bring the infinite well energy formula into play. The energy for an infinite well is given by the formula \( E = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \) . Substituting \( L = 2 \sqrt{2E/ \kappa } \) into the formula, and noting that \( E \) is on both sides, some cancellations will lead to a formula expressing \(E\).

Account for Potential Energy

The final step is to include the potential energy. It is given that the potential energy is roughly equal to the kinetic energy. Since the total energy is \( E \), each form of energy could be approximately \(E/2\). So, we can equate the kinetic energy \(E/2\) to the formula we obtained in previous step and solve for \(E\).

Key concepts

Infinite Potential Well

An infinite potential well is a foundational model in quantum mechanics, pivotal to understanding particle behavior at a quantum level. It represents an idealized system wherein a particle is confined to move within a rigid, impenetrable boundary, with infinite potential outside and zero potential inside. The beauty of this model lies in its simplicity, allowing us to explore the quantum regime free of complex interactions.

Imagine a particle, like an electron, trapped inside an infinitely deep pit with perfectly vertical walls. The particle can move freely inside this well but cannot escape it. In classical mechanics, a ball moving within a bounded space can have any energy corresponding to its motion. However, in the realm of quantum mechanics, the story is quite different. The energy of our quantum particle can only take on certain discrete values, a fundamental feature known as quantization.

The mathematics behind this involves solving the Schrödinger equation for the well, which results in wave functions that describe the probability of finding the particle at a particular position. The wave functions are akin to standing waves, which can only form at specific frequencies, corresponding to the quantized energy levels the particle can occupy. Hence, unlike a classical ball, our quantum particle can only have specified energies, a consequence of its wave-like nature confined in space.

Quantized Energies

Quantized energies are at the heart of quantum mechanics, revealing a stark contrast with the continuous energy we encounter in the everyday macroscopic world. Just as a guitar string can only vibrate at certain frequencies, particles in quantum systems such as electrons in an atom or a particle in an infinite potential well can only possess discrete energy levels.

This concept is not just a theoretical abstraction but has profound practical implications. It explains, for instance, why atoms emit and absorb light at characteristic frequencies, leading to the distinct color lines we see in spectroscopy. Each frequency corresponds to an electron jumping between fixed energy levels within the atom.

In the context of the infinite potential well, this quantization implies that a particle confined within it cannot have a smooth range of kinetic energies. Instead, its kinetic energy follows a specific set of values determined by quantum numbers and the properties of the well itself. The relationship draws upon the wave nature of particles – only certain wavelengths fit within the confines of the well, producing the allowed standing wave patterns. Each of these patterns corresponds to a quantized energy level of the particle, thus making the energy levels fixed and countable, much like the distinct rungs of a ladder.

Classical Oscillator

The classical oscillator concept illustrates a system where a particle or a body moves back and forth about a mean position, often harmonically. This model is epitomized by the iconic mass-on-a-spring, undergoing simple harmonic motion. At any given moment, the total energy of the oscillator is a mix of kinetic energy (energy due to motion) and potential energy (energy stored in the spring).

In an ideal simple harmonic oscillator, energy conservation dictates that the maximum potential energy equals the maximum kinetic energy at different points in the cycle. At the turning points, the entire energy is potential, and the kinetic energy is zero as the movement momentarily ceases before reversing direction. Conversely, at the mean position, kinetic energy peaks while potential energy is minimal.

Remarkably, when you transition from classical to quantum versions of the oscillator, you encounter quantized energy states as with the infinite potential well. But unlike the sudden drops and peaks in potential energy within the infinite well, the quantum harmonic oscillator's potential energy varies smoothly with position. It's a gentle parabolic curve rather than a steep cliff. However, even in this smoother landscape, a quantum oscillator's energies are also quantized, showcasing a universal principle of discreteness in the quantum realm that governs all confined systems, whether they are bounded by steep walls or gentle slopes.