Brian Bi

What is Schrodinger's equation?

The Schrödinger equation is an equation governing the time evolution of the wave function of a single nonrelativistic particle. A rough analogy is that the Schrödinger equation does for a quantum-mechanical particle what Newton's Second Law does for a classical particle. We can solve the Schrödinger equation to determine how a particle evolves in time, just as we use Newton's Second Law to solve for future position and momentum of a classical particle.

While Newton's Second Law precisely quantifies change in momentum, a different approach is required in quantum mechanics because particles do not have precise positions and momenta. Instead, when we measure the position of a particle, the result is inherently uncertain. There is a probability distribution on the possible results of the measurement. The same is true if we measure the momentum of a particle. From this it is clear that we can't describe the state of a quantum-mechanical particle using a set of six real numbers, the way we do in classical mechanics. To fully specify the state of a quantum-mechanical particle, we use a wave function. A wave function is a function that assigns a complex number to each point in space. So while only six numbers suffice to describe a classical particle, you need an infinite number of numbers in order to describe a quantum-mechanical particle.

When the position of a particle is measured, we are most likely to find the particle in regions of space where the magnitude of the wave function is large. To be precise, the probability density of finding the particle at a given point is given by the squared magnitude of the wave function at that point.

More on wave functions here: What is a wave function?

The Schrödinger equation is an equation in terms of the wave function of a particle. When the equation is solved, we can determine what the wave function of the particle will look like at a future time, therefore we can determine the distribution of the particle's position, momentum, and so on at future times. In classical mechanics we solve for position and momentum as functions of time, but in quantum mechanics we have to solve for a wave function, which is already a function of three coordinates, as a function of time; therefore the Schrödinger equation is a partial differential equation.

Now, Newton's Second Law $\mathbf{F} = \mathrm{d}\mathbf{p}/\mathrm{d}t$ is not enough by itself for you to work out the future motion of a particle. You have to know the formula for the force $\mathbf{F}$. For example a charged particle's motion will be different in an electric field than in a magnetic field because the two kinds of fields exert different forces. In the same way, the Schrödinger equation by itself is not enough to solve for the future wave function of a particle. You must also know the nature of the system. It may contain gravitational, electromagnetic, or other kinds of forces that influence the particle's motion.

The equation itself is usually stated in the following form:

$H\psi = i\hbar \frac{\partial \psi}{\partial t}$

Here $i = \sqrt{-1}$ and $\hbar$ is a fundamental physical constant with the dimensions of action; its value is approximately $1.05 \times 10^{-34}$ joule seconds. $H$ is a differential operator that acts on $\psi$.

If you rearrange this equation, you get
$\frac{\partial \psi}{\partial t} = -\frac{i}{\hbar} H\psi$

On the left is the time derivative of the wave function, and on the right is an operator that acts on the current value of the wave function. So, given the value of $\psi$ at an initial time, we can solve this differential equation to obtain $\psi$ at future times provided that $H$ is given. $H$ must be provided, just as $\mathbf{F}$ must be provided in Newtonian mechanics.

It might occur to you that this equation is something of a tautology, because you can study a system, determine experimentally how a system evolves in time, and then write down an $H$ operator (called the Hamiltonian) that agrees with your observations. Therefore the Schrödinger equation itself has no predictive power; it tells you no information at all unless you already know $H$ or you have a guess about the form that $H$ should take. You might say that the Schrödinger equation is really nothing more than the definition of the Hamiltonian. Again this is just like Newton's Second Law. You observe the motion of the particle and then you write down the force law; Newton's Second Law is the definition of force.

In fact the equation above is more general than the Schrödinger equation. The term "Schrödinger equation" is only used for certain forms of the above equation that resemble Newtonian mechanics. Specifically, this is what  the Schrödinger equation really looks like:

$\frac{1}{2m} P^2\psi = T\psi$

That is, it states that the squared kinetic momentum of the particle divided by twice the mass of the particle equals the kinetic energy of the particle. This is like Newtonian mechanics, but let me reiterate that the kinetic energy is no longer a real number, and the momentum is no longer a vector of three real numbers. Both are now differential operators acting upon the wave function $\psi$. Again, I reiterate that this equation is therefore not a simple algebraic equation, but a partial differential equation. Note that from this form of the Schrödinger equation we can also easily see why it only describes a non-relativistic particle. For a relativistic particle we should not expect this relation to hold between the momentum and the energy. On the other hand, the form with $H$ given previously can be used even for relativistic particles.

Here we must know the forms of the operators $P$ and $T$, which can usually be obtained by analogy with their classical forms. For a particle subject to a potential $V(x, y, z)$, for example, we have

$T = H - V$
$\mathbf{P} = \mathbf{p}$

The first equation says that the kinetic energy is the Hamiltonian minus the potential energy. (Note: the Hamiltonian represents total energy.) The second equation says that the kinetic momentum is just the total momentum. Recall that $H = i\hbar \partial/\partial t$. We can also directly write down the momentum operator $\mathbf{p}$: it is given by the formula

$\mathbf{p} = -i\hbar \nabla$

Therefore, substituting the appropriate forms for $T$ and $\mathbf{P}$, we obtain the following form for the Schrödinger equation:

$-\frac{\hbar^2}{2m} \nabla^2 \psi = i\hbar\frac{\partial\psi}{\partial t} - V\psi$

which is the form that is usually first introduced in intro physics textbooks and physical chemistry textbooks, except with the $V\psi$ term on the other side.

So once the potential $V$ and the initial conditions are given, we can solve for $\psi$ as a function of time (as well as the three spatial coordinates). $V$ might for example be an electrostatic potential of an electron bound to a proton at the origin, $V = \frac{e^2}{4\pi\epsilon_0 r}$. Solving the Schrödinger equation with this $V$ tells you the dynamics of the hydrogen atom.

In a more general electromagnetic field, we classically have the following relations
$T = H - q\phi$
$\mathbf{P} = \mathbf{p} - q\mathbf{A}$

so here we can guess that these relations also hold quantum-mechanically for the corresponding operators, giving a Schrödinger equation of the form
$\frac{1}{2m}(\mathbf{p} - q\mathbf{A})^2 \psi + q\phi \psi = i\hbar \frac{\partial \psi}{\partial t}$

where again recall that $\mathbf{p}$ stands for $-i\hbar\nabla$. This, indeed, turns out to be the correct form.

In general the Schrödinger equation is second-order in space derivatives and first-order in time derivatives. It might therefore seem that it should be similar to a heat or diffusion equation. In fact this is not the case; the Schrödinger equation is a wave equation and its solutions include propagating waves; were this not so then we wouldn't use the general name "wave functions". The reason for this is that the time derivative carries a factor of $i$. Of course, the precise details of this wave propagation depend on the precise form of the equation.

The Schrödinger equation can actually be extended to multiple particles, in the form
$\frac{P_1^2}{2m_1} + \ldots + \frac{P_N^2}{2m_N} = T$
where $T$ is the total kinetic energy operator and $P_i$ is the kinetic momentum operator for the $i$th particle. These operators all act on the "joint" wave function of all the particles, which is a function of the $3N$ coordinates. However, the Schrödinger equation cannot describe:
• Systems in which the number of particles varies. For example it cannot describe the emission or absorption of a photon by an electron in an atom, although such systems can be treated "semi-classically" by writing down
the Schrödinger equation for the electron alone and treating the electromagnetic field classically.
• Systems in which particles are relativistic. Remember, even for relativistic classical particles the relation $T = P^2/(2m)$ does not hold.
• Particles with spin. In fact, it is not too hard to modify the Schrödinger equation so that it describes particles with spin, but note that the spin can also evolve, so we have to add a term related to the angular momentum of the particle. The resulting kind of equation is no longer called the Schrödinger equation.