Physics is effectively understanding the natural world through mathematics – it’s been a highly productive and successful marriage between an abstract realm and the physical world.

Physics is almost defined by the equations that have been generated over the generations since the times of Galileo, Kepler and Newton. Examples include Maxwell’s equations, Einstein’s field equations, Einstein’s famous E=mc

^{2}equation and Boltzmann’s entropy equation. This is not an exhaustive list but it covers everything from electromagnetic radiation to gravity to nuclear physics to thermodynamics.

It is difficult to understand physics without a grasp of the mathematics, and this is true in all of the above examples. But perhaps the most difficult of all are the mathematics associated with quantum mechanics. This post is not an attempt to provide a definitive understanding but to give a very basic exposition on one of the foundational equations in the field. In so doing, I will attempt to explain its context as well as its components.

There are 3 fundamental equations associated with quantum mechanics: Planck’s equation, Heisenberg’s uncertainty principle and Schrodinger’s equation. Of course, there are many other equations involved, including Dirac’s equation (built on Schrodinger’s equation) and the QED equations developed by Feynman, Schwinger, Tomonaga and Dyson, but I’ll stop at Schrodinger’s because it pretty well encapsulates quantum phenomena both conceptually and physically.

The 3 equations are:

E =

*hf*

The first equation is simply that the energy, E, of a photon is Planck’s constant (h = 6.6 x 10

^{-34}) times its frequency,

*f*.

This is the equation that gives the photoelectric effect, as described by Einstein, and gave rise to the concept of the photon: a particle of light. The energy that a photon gives to an electron (to allow it to escape from a metal surface) is dependent on its frequency and not its intensity. The higher the frequency the more energy it has and it must reach a threshold frequency before it affects the electron. Making the photons more intense (more of them) won’t have any effect if the frequency is not high enough. Because one photon effectively boots out one electron, Einstein realised that the photon behaves like a particle and not a wave.

The second equation involves

*(called h bar) and is*~~h~~

*h*divided by 2π.

*is more commonly used in lieu of*~~h~~

*h*and it features prominently in Schrodinger’s equation.

(For future reference there is a relationship between

*f*and

*w*whereby

*w*=

*f*x 2π, which is the wave number equals frequency times 2π. This means that E =

*hf*=

*hw*/2π and becomes E =

~~h~~

*w*.)

The second equation entails Heisenberg’s uncertainty principle, which states mathematically that there are limits to what we can know about a particle’s position or its momentum. The more precisely we know its position the less precisely we know its momentum, and this equation via Planck’s constant defines the limits of that information. We know that in practice this principle does apply exactly as it’s formulated. It can also be written in terms of E and

*t*(Energy and time). This allows a virtual particle to be produced of a specific energy, providing the time duration allows it within the limits determined by Planck’s constant (it’s effectively the same equation only one uses E and

*t*in lieu of

*p*and

*x*). This has been demonstrated innumerable times in particle accelerators.

To return to Schrodinger’s equation, there are many ways to express it but I chose the following because it’s relatively easy to follow.

The first thing to understand about equations in general is that all the terms have to be of the same stuff. You can’t add velocity to distance or velocity to acceleration; you can only add (or deduct) velocities with velocity. In the above equation all the terms are Energy times a Wave function (called psi).

The terms on the right hand side are called a Hamiltonian and it gives the total energy, which is kinetic energy plus potential energy (ignoring, for the time being, the wave function).

If you have a mass that’s falling in gravity, at any point in time its energy is the potential energy plus its kinetic energy. As it falls the kinetic energy increases and the potential energy decreases, but the total energy remains the same. This is exactly what the Schrodinger equation entails. The Hamiltonian on the right gives the total energy and the term on the left hand side gives the energy of the particle (say, an electron) at any point in time via its wave function.

Another way of formulating the same equation with some definition of terms is as follows:

The Laplacian operator just allows you to apply the equation in 3 dimensions. If one considers the equation as only applying in one dimension (

*x*) then this can be ignored for the sake of explication.

Before I explain any other terms, I think it helps to provide a bit of contextual history. Heisenberg had already come up with a mathematical methodology to determine quantum properties of a particle (in this case, an electron) using matrices. Whilst it gave the right results, the execution was longwinded (Wolfgang Pauli produced 40 pages to deduce the ‘simple’ energy levels of the hydrogen atom using Heisenberg’s matrices) and Schrodinger was 'repelled' by it. An erudite account of their professional and philosophical rivalry can be found in Arthur I. Miller’s account,

*Erotica, Aesthetics and Schrodinger’s Wave Equation*, in Graham Farmelo’s excellent book,

*It Must be Beautiful; Great Equations of Modern Science*.

Schrodinger was inspired by Louis de Broglie’s insight that electrons could be described as a wave in the same way that photons could be described as particles. De Broglie understood the complementarity inherent between waves and particles applied to particles as well as light. Einstein famously commented that de Broglie ‘has lifted a corner of the great veil’.

But Schrodinger wanted to express the wave as a continuous function, which is counter to the understanding of quantum phenomena at the time, and this became one of the bones of contention between himself and Heisenberg.

Specifically, by taking this approach, Schrodinger wanted to relate the wave function back to classical physics. But, in so doing, he only served to highlight the very real discontinuity between classical physics and quantum mechanics that Heisenberg had already demonstrated. From Miller’s account (referenced above) Schrodinger despaired over this apparent failure, yet his equation became the centre piece of quantum theory.

Getting back to Schrodinger’s equation, the 2 terms I will focus on are the left hand term and the kinetic energy term on the right hand side. V (the potential energy) is a term that is not deconstructed.

The kinetic energy term is the easiest to grasp because we can partly derive it from Newtonian mechanics, in spite of the

*term.*~~h~~

In Newtonian classical physics we know that (kinetic energy) E = ½ mv

^{2}

We also know that (momentum) p = mv

It is easy to see that p

^{2}= (mv)

^{2}therefore E = p

^{2}/2m

In quantum wave mechanics p

_{x}= -i

*d/dx (I derive this separately in*~~h~~

**Addendum 5**below)

(Remember (–i)

^{2}= -1 = i

^{2}because -1 x -1 = 1)

So p

_{x}

^{2}= -

~~h~~

^{2}d

^{2}/dx

^{2}therefore E = -

~~h~~

^{2}/2m d

^{2}/dx

^{2}

Which is the kinetic energy term on the right hand side of Schrodinger’s equation (without the Laplacian operator).

I apologise for glossing over the differential calculus, but it's in another post for those interested (see

**Addendum 5**).

The term on the left hand side is the key to Schrodinger’s equation because it gives the wave function in time, which was what Schrodinger was trying to derive.

But to understand it one must employ Euler’s famous equation, which exploits complex algebra. In classical physics, wave equations do not use complex algebra (using the imaginary number,

*i*). I will return to a discourse on imaginary numbers and their specific role in quantum mechanics at the end.

e

^{ix}= cosx +

*i*sinx

This equation allows one to convert from Cartesian co-ordinates to polar co-ordinates and back, only the

*y*axis one finds in Cartesian co-ordinates is replaced by the

*i*axis and the corresponding diagram is called an Argand diagram.

In Schrodinger’s equation the wave function is expressed thus

Ψ(x, t) = Ae

^{i(kx−ωt)}where A is the wave amplitude.

If one differentiates this equation, wrt (with respect to) the term

*t*, we get the left hand term in his equation.

Differentiating an exponential function (to base

*e*) gives the exponential function and differentiating i(kx-wt) wrt

*t*gives -

*iw*. So the complete differentiated equation becomes

∂Ψ/∂t = −iωΨ

Multiplying both sides by i

*gives i*~~h~~

*∂Ψ/∂t =*~~h~~

*ωΨ*~~h~~

But from much earlier I foreshadowed that

*ω = E*~~h~~

So i

*∂Ψ/∂t =*~~h~~

*ωΨ = EΨ*~~h~~

This gives the left hand term for the famous time dependent Schrodinger wave equation.

The simplest expression is given thus:

Where H is simply the Hamiltonian.

Going back to the classical wave equation, which Schrodinger was attempting to emulate in quantum mechanics, a time dependent equation would give the position of the particle at a particular point in time, knowing what its energy would be from the Hamiltonian. However, in quantum mechanics this is not possible, and Heisenberg pointed out (according to Miller cited above) that Schrodinger’s equation did not give a position of electrons in orbits or anywhere else. However, Max Born demonstrated, by taking the modulus of the wave function (effectively the amplitude) and squaring it, you could get the probability of the position and this prediction matched experimental results.

This outcome was completely consistent with Heisenberg’s uncertainty principle which stated that determining the particle’s precise position given its momentum, which can be derived from its energy, is not possible. Schrodinger also demonstrated that his equation was mathematically equivalent to Heisenberg’s matrices.

So Schrodinger’s equation effectively didn’t tell us anything new but it became the equation of choice because it was conceptually and mathematically simpler to implement than Heisenberg’s, plus it became the basis of Dirac’s equation that was the next step in the evolvement of quantum mechanical physics.

Back in the 1920s when this was happening, there were effectively 2 camps concerning quantum mechanics: one was led by Bohr and Heisenberg and the other was led by Einstein, Schrodinger and de Broglie. Bohr developed his Copenhagen interpretation and that is effectively the standard view of quantum mechanics today. Louisa Gilder wrote an excellent book on that history, called

*The Age of Entanglement*, which I reviewed in January 2010, so I won’t revisit it here.

However, Schrodinger’s wave equation is a continuous function and therein lies a paradox, because all quantum phenomena are discrete.

In my last post (on cosmology) I referenced

*MATHS 1001*by Richard Elwes and he sums it up best:

*The basic principle is that the wave function Ψ permeates all of space and evolves according to Schrodinger’s equation. The function Ψ encodes the probability of finding the particle within any given region (as well as probabilities for its momentum, energy and so on). This theory can predict the outcomes of experimental observation with impressive accuracy.*

As Elwes then points out, once an observation is made then the particle is located and all the other probabilities become instantly zero. This is the paradox at the heart of quantum mechanics and it is entailed in Schrodinger’s equation.

His wave function is both continuous and ‘permeates all space’ but once a ‘measurement’ or ‘observation’ is made the wave function ‘collapses’ or ‘decoheres’ into classical physics. Prior to this ‘decoherence’ or ‘collapse’ Schrodinger’s wave function gives us only probabilities, albeit accurate ones.

Schrodinger himself, from correspondence he had with Einstein, created the famous Schrodinger’s Cat thought experiment to try and illustrate the philosophical consequences of this so-called ‘collapse’ of the wave function.

Equations for quantum mechanics can only be expressed in complex algebra (involving the imaginary number,

*i*) which is a distinct mathematical departure from classical physics. Again, referring to Elwes book, this number

*i*opened up a whole new world of mathematics and many mathematical methods were facilitated by it, including Fourier analysis, which allows any periodic phenomenon to be modelled by an infinite series of trigonometric functions. This leads to the Fourier transform which has application to quantum mechanics. Effectively, the Fourier transform, via an integral, allows one to derive a function for

*t*by integrating for

*dx*and finding

*x*by integrating for

*dt*. To quote Elwes again: ‘revealing a deep symmetry… which was not observable before.’

But

*i*itself is an enigma, because you can’t count an

*i*number of items the way you can with Real numbers.

*i*gives roots to polynomials that don’t appear on the Real plane. On an Argand diagram, the

*i*axes (+ and -) are orthogonal to the Real number plane. To quote Elwes: ‘…our human minds are incapable of visualizing the 4-dimensional graph that a complex function demands.’ This seems quite apt though in the world of quantum phenomena where the wave function of Schrodinger’s equation ‘permeates all space’ and cannot be determined in the classical physical world prior to a ‘measurement’. However, Born showed that by taking the modulus of the wave function and squaring it, we rid ourselves of the imaginary number component and find a probability for its existence in the physical world.

In light of this, I will give Elwes the final word on Schrodinger’s equation:

*The Schrodinger equation is not limited to the wave functions of single particles, but governs those of larger systems too, including potentially the wave function of the entire universe.*

P.S. Source material that I found useful.

**Addendum 1:**The next post furthers the discussion on this topic (without equations).

**Addendum 2:**John D. Barrow in his book,

*The Book of Universes*(see previous post) referred to Schrodinger's equation as '...the most important equation in all of mathematical physics.'

**Addendum 3:**I've written a post on complex algebra and Euler's equation here.

**Addendum 4:**According to John Gribbin in

*Erwin Schrodinger and the Quantum Revolution*, Schrodinger published a paper in 1931, where he explains Born’s contribution as multiplying the complex wave function modulus, x+iy, by its conjugate, x-iy, as multiplying the wave function in forward time by the wave function in reverse time, to obtain a probability of its position (Gribbin, Bantam Press, 2012, hardcover edition, p.161). Multiplying complex conjugates is explained in the link in

**Addendum 3**above.

**Addendum 5 (how to derive quantum momentum,**

**p**

_{x}):Basically the wave function, which exploits Euler’s famous equation, using complex algebra (imaginary numbers) is expressed thus: Ψ = Ae

^{i(kx−ωt)}

If one differentiates this equation wrt x we get ikAe

^{i(kx−ωt)}, which is ikΨ. If we differentiate it again we get d

^{2}/dx

^{2}Ψ = (ik)

^{2}Ψ.

Now k is related to wavelength (λ) by 2π such that k = 2π/λ.

And from Planck’s equation (E = hf) and the fact that (for light) c = f λ we can get a relationship between momentum (p) and λ. If p = mc and E = mc

^{2}, then p = E/c. Therefore p = hf/f λ which gives

**p = h/λ**, effectively the momentum version of Planck’s equation. Note that p is related to wavelength (space) and E is related to frequency (time).

This then is the quantum equation for momentum based on h (Planck’s constant) and λ. And, of course, according to Louis de Broglie, particles as well as light can have wavelengths.

And if we substitute 2π/k for λ we get p = hk/2π which can be reformulated as

k = p/

And substituting this in (ik)

^{2}we get –(p/

^{2}{ i

^{2}= -1}

So d

^{2}/dx

^{2 }Ψ = -(px/

^{2}Ψ or p

_{x}

^{2}= -

^{2}d

^{2}/dx

^{2}(which is inserted into the Time Dependent Schrodinger Equation, above).