A free proton has a wave function given by \(\Psi(x, t)=A e^{i\left(5.02 \times 10^{11} x-8.00 \times 10^{15} t\right)}\) The coefficient of \(x\) is inverse meters \(\left(\mathrm{m}^{-1}\right)\) and the coefficient on \(t\) is inverse seconds \(\left(\mathrm{s}^{-1}\right) .\) Find its momentum and energy.

The wave function, often represented as \( \Psi(x, t) \), is a fundamental concept in quantum mechanics that encapsulates all the information about a quantum system's state. Imagining as the DNA of a particle, it provides a comprehensive description of the particle's behavior, such as its location and momentum probability distributions.

For example, a wave function like \( \Psi(x, t) = A e^{i(5.02 \times 10^{11} x - 8.00 \times 10^{15} t)} \) can seem mystifying at first glance. But let's decode it: \(|\Psi(x, t)|^2\), the square of the wave function's magnitude, tells us the probability density of finding the particle at a particular position and time. The expression \(|A|^2\) sees 'A' as the amplitude, which could be associated with the probability density when we square it.

In our exercise, the complex exponential component manifests the wave-like nature of quantum particles, also connecting the probabilities with the nicely predictable patterns of waves. This ties back to the fact that particles at the quantum level behave like both particles and waves, an idea at the heart of wave-particle duality.

The de Broglie wavelength, denoted as \( \lambda \), is another revolutionary concept in quantum mechanics introduced by Louis de Broglie. This idea is a shining example of wave-particle duality. It proposes that particles can exhibit wave-like properties and that each particle has a wavelength inversely proportional to its momentum.

Using the formula \( p = \frac{h}{\lambda} \), where 'p' is the momentum and 'h' is Planck's constant, we can determine a particle's momentum given its wavelength. In the case of our exercise, the wavenumber 'k', taken from the coefficient of 'x' in the wave function, is related to the de Broglie wavelength by \( k = \frac{2\pi}{\lambda} \). Therefore, once we determine \( \lambda \), we can calculate the particle's momentum, a bridge between wave-like and particle-like properties, showcasing the versatility of quantum mechanics in describing the microcosmic world.

The Planck-Einstein relation is a pivotal equation in quantum mechanics linking the energy of photons with their frequency, using the equation \( E = hf \), where 'E' is energy, 'h' is Planck's constant, and 'f' is the frequency of the wave.

In the realm of quantum mechanics, where quantization is king, this formula implies that energy comes in discrete packets, called quanta. This discrete nature demarks a clear departure from classical mechanics where energy is seen as a continuous flow. For our proton exercise, the angular frequency \( \omega \) is directly related to the frequency by \( \omega = 2\pi f \) and is thus critical for calculating the energy of the particle.

Moreover, the Planck-Einstein relation embodies the quantization of electromagnetic waves, a key principle that underlies the understanding of phenomena such as the photoelectric effect and Planck's black body radiation law. It's through applying this relation that we can unfold the quantum world's intricacies, like determining the energy of a particle represented in wave function, and appreciate the quantized harmony of the universe's most fundamental levels.