A particle with mass \(m\) is described by the following wave function: \(\psi(x)=A \cos k x+B \sin k x,\) where \(A, B\) and \(k\) are constants. Assuming that the particle is free, show that this function is the solution of the stationary Schrödinger equation for this particle and find the energy that the particle has in this state.

The wave function, often denoted as \( \psi(x) \), is a fundamental concept in quantum mechanics. It is a mathematical function that describes the quantum state of a particle or system of particles. More than just a simple equation, the wave function encapsulates all the information about a system's properties and its behavior.

For a free particle, the wave function takes the form of a combination of sine and cosine terms, like the given \( \psi(x)=A \cos kx + B \sin kx \), relevant to our exercise. Understanding wave functions is crucial because they allow us to predict the probability of finding a particle at a particular location in space. This probabilistic nature is one of the core tenets of quantum mechanics, moving away from the deterministic views of classical physics.

However, to interpret what the wave function signifies, we utilize the principle that the square of the wave function's magnitude, \( |\psi(x)|^2 \), represents the probability density. Moreover, for a wave function to be physically acceptable, it must be normalized, meaning that the total probability of finding a particle must be one when considering all possible positions.

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It provides a mathematical framework for understanding the properties and behavior of matter and energy at this small scale, which often defies the principles observed in classical mechanics.

At the heart of quantum mechanics lies the Schrödinger equation, a key piece in solving quantum problems that dictates how the wave function evolves over space and time. One form of the Schrödinger equation applied in our exercise is the stationary or time-independent Schrödinger equation which describes systems where the energy levels are quantized and constant in time. It is this equation that allows us to find allowable energy levels and wave functions for quantum systems like the particle described in the given exercise.

Solving these equations isn't just mathematical routine; it provides insights into the behavior of particles. For example, it shows us that particles like electrons do not occupy specific orbits but rather exist in areas known as 'orbitals', a concept that arises directly from the solutions to these quantum equations.

When discussing particle energy within the realm of quantum mechanics, we are often referring to the energy associated with a particle's quantum state. The energy is not arbitrary; quantization is a powerful principle here, which means that particles can only have certain discrete energy values.

In our exercise, after confirming the given wave function is a solution to the time-independent Schrödinger equation, we determined the associated energy using the wave number \(k\). The relationship \(E = \frac{\hbar^2 k^2}{2m}\) reveals that energy depends on the square of the wave number and inversely on the mass of the particle. This is an example of kinetic energy at the quantum scale, since in the absence of potential energy (as for a free particle), all the particle's energy is kinetic.

The concept of particle energy in quantum mechanics further illustrates the departure from classical views, as energy levels result from the wave-like properties of particles, wave functions, and the constraints imposed by boundary conditions or potentials. This intrinsic energy influences how particles interact, bond, decay, and essentially participate in the processes fundamental to the natural world.