Question

Write out the total wave function \(\Psi(x, t)\) for an electron in the \(n=3\) state of a \(10 \mathrm{nm}\) wide infinite well. Other than the symbols \(x\) and \(t,\) the function should include only numerical values.

Short answer

\(\Psi(x,t) = \sqrt{\frac{2}{10^{-9}}}\sin \left(\frac{3\pi x} {10^{-9}} \right) \cdot e^{(-i \frac{9 \pi^2 (1.0545718 \times 10^{-34}) t}{2 * (9.10938356\times10^{-31}) * (10^{-9})^2})} \)

Step-by-step solution

Get Spatial Component

Substitute the values into the spatial function as follows: \(\psi(x)=\sqrt{\frac{2}{10^{-9}}}\sin \left(\frac{3\pi x} {10^{-9}} \right)\)

Get Temporal Component

Now, substitute the values into the temporal function. Remember that \(\hbar = 1.0545718 \times 10^{-34} \mathrm{m}^2 \mathrm{kg} / \mathrm{s}\) and the mass of an electron \(m = 9.10938356\times10^{-31}\):\(\phi(t) = e^{(-i \frac{(3)^2 \pi^2 (1.0545718 \times 10^{-34}) t}{2 * (9.10938356\times10^{-31}) * (10^{-9})^2})}\)

Form Total Wave Function

Combine both spatial and temporal parts to form the complete wave function: \(\Psi(x,t) = \psi(x) \cdot \phi(t)\)

Key concepts

Quantum Mechanics

The intriguing world of quantum mechanics diverges from classical physics, particularly at the atomic and subatomic scales. This realm is governed by principles that confound our everyday experiences, operating under rules that allow particles, like electrons, to exhibit wave-like properties, and vice versa. This dual nature is captured elegantly in the concept of the wave function, a mathematical expression that summarizes all that can be known about a quantum system.

The exercise provided explores quantum mechanics by asking students to calculate the wave function for an electron in a specific quantum state within an infinite potential well. To understand this, it's vital to grasp that unlike the orbits of planets, the position and momentum of an electron cannot be pinpointed simultaneously with absolute certainty. Instead, quantum mechanics describes probable locations and momenta through complex mathematical constructs called wave functions, denoted by \( \Psi(x, t) \). These wave functions evolve over time, encapsulating the probabilistic nature of quantum systems.

Electron Probability Amplitude

A critical concept within quantum mechanics is the probability amplitude for an electron's position. This concept emphasizes that the wave function represents more than just the wave nature of particles; it also provides a way to calculate the probability of finding an electron at a specific position. The square of an electron's wave function, \( |\psi(x)|^2 \), indicates the probability density – the likelihood of locating an electron in a particular region of space.

The amplitude itself, represented by \( \psi(x) \), provides the spatial component of an electron's wave function in the step-by-step solution. When this spatial wave function is combined with its temporal counterpart, \( \phi(t) \), it defines the state of the electron over time. By following these steps, students can uncover the beautifully complex dance of probabilities that dictates where an electron might be at any given moment.

Infinite Potential Well

An infinite potential well is a quintessential concept in quantum mechanics, serving as a fundamental model for understanding the quantization of energy levels in a confined space. Imagine a particle, like an electron, trapped inside a well with perfectly rigid walls that it cannot escape. This well is a hypothetical construct used to simplify the complex reality of particle confinement.

Within this well, the particle's energy levels are discrete, meaning the electron can only occupy certain states, each corresponding to a distinct quantum number \( n \). The solution provided showcases the calculation for the \( n=3 \) state, indicating the third allowable energy state for the electron in the well. For students, visualizing these energy states can be as simple as picturing a musical instrument's strings, where each vibration mode corresponds to a different note – except in quantum mechanics, each 'note' represents a possible energy level for the particle.