Planck hypothesized in 1900 that, for an electromagnetic wave with frequency $\nu$, the only possible energies are of the form $nh\nu$, where $n$ is an integer and $h$ is a new constant. Einstein generalized it in 1905 and proposed that, light is made up of a beam of photons, each with energy $h\nu$. This allowed Einstein to explain the photoelectric effect. The photon was shown to exist 18 years later in 1923, by the Compton effect. Particle parameters and wave parameters can be related by: \[ \boxed{E = h\nu = \hbar \omega} \tag{1} \] where $\hbar = h / 2\pi$ and $h \approx 6.62 \times 10^{-34}$ Joule $\times$ second. And \[ \boxed{\mathbf{p} = \hbar \mathbf{k}} \tag{2} \] where $\mathbf k$ is the wave vector. These are the Planck-Einstein relations.

The study of atomic emission and absorption spectra uncovered a fact, an atom emits or absorbs only photons having well-determined frequencies. The emission or absorption of a photon results in a jump in the energy of the atom from one permitted value to another. This implies that the energy of the atom can take on only certain discrete values and is hence, quantized.

We begin with the simplest case of all: the free particle. When $V(\mathbf r, t) = 0$ everywhere, the Schrödinger equation becomes: \[ i\hbar \frac{\partial}{\partial t} \psi(\mathbf r, t) = - \frac{\hbar^2}{2m}\Delta\psi(\mathbf r, t) \tag{7} \] This differential equation is satisfied by solutions of the form: \[ \psi(\mathbf r, t) = A e^{i(\mathbf k \cdot \mathbf r - \omega t)} \tag{8} \] where $A$ is a constant and \[ \omega = \frac{\hbar \mathbf k^2}{2m} \tag{9} \] This wave represents a particle whose probability of presence is uniform everywhere, since \[ |\psi(\mathbf r, t)|^2 = |A|^2 \tag{10} \] We also see that, (9) implies that energy $E$ and momentum $\mathbf p$ of a free particle satisfy: \[ E = \frac{\mathbf p^2}{2m} \tag{11} \] which is well-known in classical mechanics.

We begin by noticing that (6) is linear and homogeneous in $\psi$. According to the principle of superposition, every linear combination of plane waves satisfying (9) will also be a solution of equation (7). Such a superposition can be written as: \[ \psi(\mathbf r, t) = \frac{1}{(2\pi)^{3/2}} \int g(\mathbf k) e^{i[\mathbf k \cdot \mathbf r-\omega(k)t]} \; \mathrm d^3k \tag{12} \] A wave function like this is called a three-dimensional wave packet. A plane wave of form (8), whose modulus is constant throughout all space, is not square-integrable, and hence, not normalizable. Therefore, it cannot represent a physically-realizable state. On the other hand, a superposition of plane waves like $(12)$ can be square-integrable. For simplicity, we will study the one-dimensional case: \[ \psi(x, t) = \frac{1}{(2\pi)^{1/2}} \int_{-\infty}^{\infty} g(k) e^{i[k x-\omega(k)t]} \; \mathrm dk \tag{13} \] If we choose $t = 0$, the wave function becomes: \[ \psi(x, 0) = \frac{1}{(2\pi)^{1/2}} \int_{-\infty}^{\infty} g(k) e^{ikx} \; \mathrm dk \tag{14} \] Hence, $g(k)$ is simply the Fourier transform of $\psi(x, 0)$: \[ g(k) = \frac{1}{(2\pi)^{1/2}} \int_{-\infty}^{\infty} \psi(x, 0) e^{-ikx} \; \mathrm dx \tag{15} \]

The wave function of a particle when $V(\mathbf r)$ is time-independent must satisfy: \[ i\hbar \frac{\partial}{\partial t} \psi(\mathbf r, t) = - \frac{\hbar^2}{2m}\Delta\psi(\mathbf r, t) + V(\mathbf r)\psi(\mathbf r, t) \tag{16} \] In this case, the Schrödinger equation can be solved by separation of variables. We will look for solutions of the form: \begin{align} \psi(\mathbf r, t) & = \varphi(\mathbf r) \phi(t) \tag{17} \\ \\ \implies i\hbar \, \varphi(\mathbf r) \frac{\mathrm d \phi(t)}{\mathrm d t} & = - \frac{\hbar^2}{2m}\Delta\varphi(\mathbf r) \phi(t) + V(\mathbf r)\varphi(\mathbf r) \phi(t) \tag{18} \\ \\ \implies \frac{i\hbar}{\phi(t)} \frac{\mathrm d \phi(t)}{\mathrm d t} & = - \frac{1}{\varphi(\mathbf r)}\frac{\hbar^2}{2m}\Delta\varphi(\mathbf r) + V(\mathbf r) \tag{19} \\ \end{align} This equation equates a function of $t$ only with a function of $\mathbf r$ only. Hence, each of these functions is a constant, which we will set equal to $\hbar \omega$. The left-hand side gives us: \[ \phi(t) = Ae^{-i\omega t} \tag{20} \] And, the right-hand side: \[ \left[- \frac{\hbar^2}{2m}\Delta + V(\mathbf r)\right]\varphi(\mathbf r) = \hbar \omega \, \varphi(\mathbf r) \tag{21} \] We will absorb $A$ in $\varphi(\mathbf r)$: \[ \psi(\mathbf r, t) = \varphi(\mathbf r)e^{-i\omega t} \tag{22} \] This is called a stationary solution of the Schrödinger equation. A stationary state is a state with a well-defined energy $E = \hbar\omega$. Hence, we can re-write $(21)$ as: \[ \boxed{H \, \varphi(\mathbf r) = E \, \varphi(\mathbf r)} \tag{23} \] where $H$ is the differential operator: \[ H = - \frac{\hbar^2}{2m}\Delta + V(\mathbf r) \tag{24} \] $H$ is a linear operator and $(23)$ is the eigenvalue equation of $H$. The allowed energies are the eigenvalues of $H$. Equation $(23)$ is called the time-independent Schrödinger equation. It enables us to find, amongst all the possible states of the particle, those which are stationary. We will label the various possible values of the energy $E$ and the corresponding eigenfunctions $\varphi(\mathbf r)$ with index $n$. Thus: \[ H \, \varphi_n(\mathbf r) = E_n \, \varphi_n(\mathbf r) \tag{25} \] And, wave functions of the stationary states of the particle: \[ \psi_n(\mathbf r, t) = \varphi_n(\mathbf r)e^{-iE_nt/\hbar} \tag{26} \] The (time-dependent) Schrödinger equation $(6)$ has a property that every linear combination of solutions is itself a solution: \[ \psi(\mathbf r, t) = \sum_n c_n\varphi_n(\mathbf r)e^{-iE_nt/\hbar} \tag{27} \] where the coefficients $c_n$ are complex constants. In particular, we have: \[ \psi(\mathbf r, 0) = \sum_n c_n\varphi_n(\mathbf r) \tag{28} \] If we know $\psi(\mathbf r, 0)$, the state of the particle at $t = 0$. We can always decompose it in terms of eigenfunctions of $H$. The corresponding solution $\psi(\mathbf r, t)$ of the Schrödinger equation is then given by $(27)$, by simply multiplying each term of $(28)$ by $e^{-iE_nt/\hbar}$ where $E_n$ is the eigenvalue associated with $\varphi_n(\mathbf r)$.