Schrodinger's Equation: Explained & How to Use It

The Schrodinger equation is the most fundamental equation in quantum mechanics, and learning how to use it and what it means is essential for any budding physicist. The equation is named after Erwin Schrödinger, who won the Nobel Prize along with Paul Dirac in 1933 for their contributions to quantum physics.

Schrodinger's equation describes the wave function of a quantum mechanical system, which gives probabilistic information about the location of a particle and other observable quantities such as its momentum. The most important thing you’ll realize about quantum mechanics after learning about the equation is that the laws in the quantum realm are ​very different​ from those of classical mechanics.

The Wave Function

The wave function is one of the most important concepts in quantum mechanics, because every particle is represented by a wave function. It is typically given the Greek letter psi (​Ψ​), and it depends on position and time. When you have an expression for the wave function of a particle, it tells you everything that can be known about the physical system, and different values for observable quantities can be obtained by applying an operator to it.

The square of the modulus of the wave function tells you the probability of finding the particle at a position ​x​ at a given time ​t​. This is only the case if the function is “normalized,” which means the sum of the square modulus over all possible locations must equal 1, i.e. that the particle is ​certain​ to be located ​somewhere​.

Note that the wave function only provides probabilistic information, and so you can’t predict the result of any one observation, although you ​can​ determine the average over many measurements.

You can use the wave function to calculate the ​“expectation value”​ for the position of the particle at time ​t​, with the expectation value being the average value of ​x​ you would obtain if you repeated the measurement many times.

Again, this doesn’t tell you anything about a particular measurement. In fact, the wave function is more of a probability distribution for a single particle than anything concrete and reliable. By using the appropriate operator, you can also obtain expectation values for momentum, energy and other observable quantities.

The Schrodinger Equation

The Schrodinger equation is linear partial differential equation that describes the evolution of a quantum state in a similar way to Newton’s laws (the second law in particular) in classical mechanics.

However, the Schrodinger equation is a wave equation for the wave function of the particle in question, and so the use of the equation to predict the future state of a system is sometimes called “wave mechanics.” The equation itself derives from the conservation of energy and is built around an operator called the Hamiltonian.

The simplest form of the Schrodinger equation to write down is:

H Ψ = iℏ \frac{\partialΨ}{\partial t}

Where ℏ is the reduced Planck’s constant (i.e. the constant divided by 2π) and ​H​ is the Hamiltonian operator, which corresponds to the sum of the potential energy and kinetic energy (total energy) of the quantum system. The Hamiltonian is a fairly long expression itself, though, so the full equation can be written as:

−\frac{ ℏ^2}{2m} \frac{\partial^2 Ψ}{\partial x^2} + V(x) Ψ == iℏ \frac{\partialΨ}{\partial t}

Noting that sometimes (for explicitly three-dimensional problems), the first partial derivative is written as the Laplacian operator ∇2. Essentially, the Hamiltonian acts on the wave function to describe it’s evolution in space and time. But in the time-independent version of the equation (i.e. when the system doesn’t depend on ​t​), the Hamiltonian gives the energy of the system.

Solving the Schrodinger equation means finding the ​quantum mechanical wave function​ that satisfies it for a particular situation.

The Time-Dependent Schrodinger Equation

The time-dependent Schrodinger equation is the version from the previous section, and it describes the evolution of the wave function for a particle in time and space. A simple case to consider is a free particle because the potential energy ​V​ = 0, and the solution takes the form of a plane wave. These solutions have the form:

Ψ = Ae^{kx −ωt}

Where ​k​ = 2π / ​λ,​ ​λ​ is the wavelength, and ​ω​ = ​E​ / ℏ.

For other situations, the potential energy part of the original equation describes boundary conditions for the spatial part of the wave function, and it is often separated into a time-evolution function and a time-independent equation.

The Time-Independent Schrodinger Equation

For static situations or solutions that form standing waves (such as the potential well, “particle in a box” style solutions), you can separate the wave function into time and space parts:

Ψ(x, t) = Ψ(x) f(t)

When you go through this in full, the time portion can be cancelled out, leaving a form of the Schrodinger equation that ​only​ depends on the position of the particle. The time independent wave function is then given by:

H Ψ(x) = E Ψ(x)

Here ​E​ is the energy of the quantum mechanical system, and ​H​ is the Hamiltonian operator. This form of the equation takes the exact form of an eigenvalue equation, with the wave function being the eigenfunction, and the energy being the eigenvalue when the Hamiltonian operator is applied to it. Expanding the Hamiltonian into a more explicit form, it can be written in full as:

−\frac{ ℏ^2}{2m} \frac{\partial^2 Ψ}{\partial x^2} + V(x) Ψ = E Ψ(x)

The time part of the equation is contained in the function:

f (t) = e^{\frac{iEt}{ℏ}}

Solutions to the Time-Independent Schrodinger Equation

The time-independent Schrodinger equation lends itself well to fairly straightforward solutions because it trims down the full form of the equation. A perfect example of this is the “particle in a box” group of solutions where the particle is assumed to be in an infinite square potential well in one dimension, so there is zero potential (i.e. ​V​ = 0) throughout, and there is no chance of the particle being found outside of the well.

There is also a finite square well, where the potential at the “walls” of the well isn’t infinite and even if it’s higher than the particle’s energy, there is ​some​ possibility of finding the particle outside it due to quantum tunneling. For the infinite potential well, the solutions take the form:

Ψ(x) = \sqrt{\frac{2}{L}} \sin \bigg(\frac{nπ}{L}x\bigg)

Where ​L​ is the length of the well.

A delta function potential is a very similar concept to the potential well, except with the width ​L​ going to zero (i.e. being infinitesimal around a single point) and the depth of the well going to infinity, while the product of the two (​U0) remains constant. In this very idealized situation, there is only one bound state, given by:

Ψ(x) = \frac{\sqrt{mU_0}}{ℏ}e^{-\frac{mU_0}{ℏ^2}\vert x\vert}

With energy:

E = - \frac{mU_0^2}{2ℏ^2}

Hydrogen Atom Solution to the Schrodinger Equation

Finally, the hydrogen atom solution has obvious applications to real-world physics, but in practice the situation for an electron around the nucleus of a hydrogen atom can be seen as pretty similar to the potential well problems. However, the situation is three-dimensional and is best described in spherical coordinates ​r​, ​θ​, ​ϕ​. The solution in this case is given by:

Ψ(x) = NR_{n,l}(r)P^m_{l}(\cos θ)e^{imϕ}

Where ​P​ are the Legendre polynomials, ​R​ are specific radial solutions, and ​N​ is a constant you fix using the fact that the wave function should be normalized. The equation yields energy levels given by:

E = - \frac{\mu Z^2e^4}{8ϵ_0h^2n^2}

Where ​Z​ here is the atomic number (so ​Z​ = 1 for a hydrogen atom), ​e​ in this case is the charge of an electron (rather than the constant ​e​ = 2.7182818...), ​ϵ0 is the permittivity of free space, and ​μ​ is the reduced mass, which is based on the masses of the proton and the electron in a hydrogen atom. This expression is good for any hydrogen-like atom, meaning any situation (including ions) where there is one electron orbiting a central nucleus.

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