Considering the failures of the Bohr model, Erwin Schrödinger and Werner Heisenberg proposed a major change in the paradigm regarding the electron. In several breakthrough papers (19251927) they attributed wave properties to electrons and they each received Nobel prizes for developing the theories of Wave Mechanics (or the "New Quantum Mechanics"). This approach treated electrons as having "dual" nature: possessing properties of both waves and particles.
Schrödinger's Equation describes the behavior of the electron (in a hydrogen atom) in three dimensions. It is a mathematical equation that defines the electron’s position, mass, total energy, and potential energy. The simplest form of the Schrödinger Equation is as follows:
\[\hat{H}\psi = E\psi \nonumber \]
where \(\hat{H}\) is the Hamiltonian operator, \(E\) is the energy of the electron, and \(\psi\) is the wavefunction.
The Hamiltonian, \(\hat{H}\)
The Hamiltonian operator is like a set of instructions that tells us what to do with the function that follows it. A Hamiltonian operator is a function over threedimensional space that corresponds to the sum of kinetic energies and potential energies of the particles in a system, one electron and its nucleus in this case. The Hamiltonian operator for a oneelectron system is:
\[\hat{H}=\dfrac{h{^2}}{8\pi{^2}m_e}\left(\dfrac{\partial{^2}}{\partial{x^2}}+\dfrac{\partial{^2}}{\partial{y^2}}+\dfrac{\partial{^2}}{\partial{z^2}}\right)\dfrac{Ze^2}{4\pi{}\epsilon_0{r}}, \nonumber \]
where \(h\) is Planck's constant, \(m_e\) is the mass of the electron, \(e\) is the charge of the electron, \(r\) is the distance from the nucleus (\(r=\sqrt{x^2+y^2+z^2}\)), \(Z\) is the charge of the nucleus, and \(4\pi{}\epsilon_0\) is the permittivity of a vacuum.
Kinetic Energy
The first part of the Hamiltonian written above, \(\dfrac{h{^2}}{8\pi{^2}m_e}\left(\dfrac{\partial{^2}}{\partial{x^2}}+\dfrac{\partial{^2}}{\partial{y^2}}+\dfrac{\partial{^2}}{\partial{z^2}}\right)\) describes the kinetic energy of the electron. This is the energy due to motion of the electron.
Potential energy
The second part written above, \(\dfrac{Ze^2}{4\pi{}\epsilon_0{r}}\), describes the potential energy of the electron, and is commonly written as \(V(r)\) or \(V(x,y,z)\).
\[V(x,y,x) = \dfrac{Ze^2}{4\pi{}\epsilon_0{r}} = \dfrac{Ze^2}{4\pi{}\epsilon_0{\sqrt{x^2+y^2+z^2}}} \nonumber \]
The potential energy depends on the attractive electrostatic force between the electron and the nucleus. You might notice that this attraction is essentially the same as the electrostatic force defined by Coulomb's law. And, just as in Coulomb's law, when two opposite charges are attracted to one another, the potential energy of the force is negative. Thus, when an electron is close to the nucleus, the potential energy is a large negative number corresponding to a strong attractive force. When an electron is farther from the nucleus, the potential energy is still negative but with a smaller magnitude, corresponding to a weaker attractive force. If the electron is very far from the nucleus (\(r = \infty\)) then the attractive force, and the potential energy, is zero.
The Wavefunction, \(\psi\)
In simple terms, the wavefunction (\(\psi\)) of an electron describes the electron's position in space, relative to the nucleus. The square of the \(\psi\) describes an atomic orbital. We can't define the position too exactly because we would violate the Heisenberg Uncertainty principle, but we can define its wave. A simple example of a \(\psi\) is described in the next section: Particle in a Box. Here, we will describe the \(\psi\) in general terms. Generally, in a oneelectron atom, the electron \(\psi\) is defined by the wave's distance from the nucleus and its angle with respect to the x, y, and z axes of the atom's Cartesian coordinates (the nucleus is at the origin). The general form of the (\(\psi\)) for an electron in a hydrogen atom can be written as follows:
\[\psi_{n,l,m_l} = R_{n,l}(r) + Y_{l,m_l}(\theta,\phi) \nonumber \]
Quantum Numbers define \(\psi\)
The (\(\psi\)) is defined by three of the quantum numbers: \(n\), \(l\), and \(m_l\). These quantum numbers will be discussed more in a later section (2.2.2) The radial variation, \(R\), depends on the electron's distance from the nucleus. The quantum numbers \(n\) (energy level) and \(l\) (orbital type) define \(R\). Since \(n\) must be an integer, there are only certain allowed values for the solution to the wavefunction.
The angular variation, \(Y\), depends on the angle with respect to the x, y, and z coordinates, and depends on the quantum numbers \(l\) (the orbital type) and \(m_l\) (the angular momentum, or the specific orbital). For example \(p_x\) lies along the \(x\) axis, while \(p_y\) points in a different direction in space.
For review, a list of the quantum numbers, their values, and meanings are in the table below.
SYMBOL 
NAME 
VALUES 
MEANING 
\(n\) 
principal 
\(1,2,3...\)(any integer) 
energy level, shell 
\(l\) 
angular momentum 
\(0 \rightarrow n1\) 
subshell, \(0=s, 1=p, 2=d, 3=f...\)
this is the angular dependence of the orbital, shape of the orbital
*letters have historical meaning, sharp, principle, diffuse, fundamental

\(m_l\) 
magnetic 
\(+l \rightarrow l\) 
orientation of angular momentum in space, orbital 
\(m_s\) 
spin 
\(+\frac{1}{2}, \frac{1}{2}\) 
the imaginary property we call "spin", up or down 
Some important considerations and limitations
Although it might seem like there could be any value of x, y, and z for the Hamiltonian, these values are limited by the allowed positions of electrons according to \(\psi\), which is limited by integer values of \(n\). In other words the allowed solutions are quantized. However, there are an infinite number of values for \(n\) from \(n=1\rightarrow\infty\), so there are also infinite solutions to the Schrödinger equation.
The \(\psi\) describes the wave properties of an electron. The probability of finding the electron somewhere in space is the square of the wavefunction (\(\psi^2\) or \(\psi \psi^*\)). In other words, \(\psi^2\) describes the shape and size of an electron's orbital (the shapes you already know).
There are some requirements for a physically realistic and meaningful solution for \(\psi\), and thus \(\psi^2\).
 There is only one possible value for \(\psi\) for any set of the three quantum numbers \(n, l, m_l\).
 The \(\psi\) approaches zero as \(r\) approaches infinity, and so \(\psi^2\) also approaches zero as \(r\rightarrow\infty\).
 The wavefunction must be normalized. In other words, the total probability of finding the electron in all of space must be 1. \[\int_{\text {all space}} \psi_{A} \psi_{A}^{*} d \tau=1 \nonumber \]
 Any two orbitals must not occupy the same space. In other words, any two orbitals in an atom are orthogonal. If \(\psi_{A}\) and \(\psi_{B}\) are wavefunctions for different orbitals in the same atom, \[\int_{\text {all space}} \psi_{A} \psi_{B}^{*} d \tau=0 \nonumber \]
 The probability of finding the electron anywhere in infinite space must be defined. This means that the wave functions and their first derivatives must be continuous (i.e. not change abruptly from one point to the next).
Awesome Sources for further reading