Schrödinger's Wave Equation

In 1926, Austrian physicist Erwin Schrödinger reasoned that if electrons behave as waves, then it should be possible to describe them using a wave equation, like the equation that describes the vibrations of strings, or Maxwell's equation for electromagnetic waves[1].

1. Wave functions

1.1 Classical wave functions

A wave equation typically describes how a wave function evolves in time. A function describes a relationship between two values. The function f(x) = x+1, for example, is a function because for every value of x you get a new value of f(x).

A wave function describes the behaviour of something that is waving. In the case of Maxwell's equations, the wave function describes the behaviour of the electric and magnetic fields. In the case of a wave on a string, the wave function describes the displacement of the string.

The wave function of a wave travelling in the x direction, with angular frequency ω = 2πν - where ν is the frequency - is:

Ψ(x,t) = Acos(kx-ωt) = Aei(kx-ωt).

Here Ψ(x,t) is the wave function, A is the amplitude, e, and i are constants, k = 2π/λ, where λ is the wavelength, x is the position, and t is time.

1.2 Quantum wave functions

Schrödinger saw that for an object with E = hν (the Planck relation, where E equals energy and h is Planck's constant), and λ = h/p (the de Broglie wavelength, where p is momentum), this equation can be rewritten as a quantum wave function:

Ψ(x,t) = Aei(kx-ωt),

using k = 2π/λ,

Ψ(x,t) = Aei(xλ-ωt),

using ω = 2πν,

Ψ(x,t) = Aei2π(xλ-νt),

using λ = h/p,

Ψ(x,t) = Aei2π(pxh-νt),

using E = hν,

Ψ(x,t) = Aei2πh(px-Et),

Finally, using ħ = h/2π gives,

Ψ(x,t) = Aei(px - Et)ħ.

This is the quantum wave function. The Schrödinger equation shows how the quantum wave function changes over time.

1.3 The Schrödinger wave equation

Schrödinger showed that position and momentum are related using calculus, a branch of mathematics developed by English natural philosopher Isaac Newton and German mathematician Gottfried Leibniz in the late 1600s.

One branch of calculus is known as differentiation, a mathematical method for measuring how a function changes as its input changes. If you differentiate position with respect to time, for example, then you are measuring velocity. If you differentiate velocity with respect to time, then you are measuring acceleration.

This method can be used in scenarios where the equations Velocity = Distance/Time and Acceleration = Velocity/Time aren't applicable, such as when the velocity or acceleration is constantly changing.

Schrödinger showed that if you differentiate Ψ(x,t) with respect to position, x, then the result is equal to Ψ(x,t) multiplied by the momentum (p), and a constant (i/ħ):

dΨ(x,t)/dx
=
ip/ħ
Ψ(x,t).

If you differentiate Ψ(x,t) with respect to time, t, then the result is equal to Ψ(x,t) multiplied by the energy (E), and a constant (-i/ħ):

dΨ(x,t)/dt
=
-iE/ħ
Ψ(x,t).

For an electron travelling through an electric field, for example, the total energy is equal to the kinetic energy plus the potential energy of the field.

The kinetic energy (K) equals
1/2
mV2, where m is mass, and V is velocity.
Using p = mV, K =
1/2
mV2 =
1/2
m2V2/m
=
p2/2m
.
The potential energy of the field equals PE(x), and so E =
p2/2m
+ PE (x).
Putting this into
dΨ(x,t)/dt
=
Eiħ
Ψ(x,t) gives iħ
dΨ(x,t)/dt
=
p2/2m
Ψ(x,t) + PE(x)Ψ(x,t).
Finally, using
d2Ψ(x,t)/dx2t
=
p2/2
Ψ(x,t) gives:
iħ
dΨ(x,t)/dt
=
2/2m
d2Ψ(x,t)/dx2
+ PE(x)Ψ(x,t),

or

EΨ(x,t) = HΨ(x,t)

This is the time-dependent Schrödinger equation - or wave equation - for a single non-relativistic charged particle moving in an electric field.

It describes all the features of the electron that we can measure, and can be extended to include any other object under almost any other force.

The Schrödinger equation can be used to make the exact same predictions as German physicist Werner Heisenberg's uncertainty principle. It can calculate where electron waves will be situated within an atom, and predict where spectral lines will occur.

Schrödinger's equation describes the world in terms of continuously evolving waves, and Heisenberg's describes it in terms of particles that undergo 'jumps' from one place to another without moving through the space in-between. Many physicists preferred Schrödinger's approach because it was easier to visualise and used more familiar mathematics.

Schrödinger went on to show that his wave equation is equivalent to Heisenberg's uncertainty principle[2], although they both argued for the superiority of their own approach[3]. Danish physicist Niels Bohr however, believed that both views were equally valid[4].

1.4 Probability clouds and the Born rule

In classical wave equations, the wave function has a real meaning, it describes something that is physically waving, but Schrödinger's wave equation had no physical interpretation.

In the 1960s, American physicist Richard Feynman stated:

"Where did we get that [equation] from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger"[5].

In 1926, Schrödinger believed that electron waves were always spread out across all of space, and that the square of the wave function gave the charge density of the electron wave in any particular location[6][7a].

This was a reasonable assumption since the wave appeared to be densest in the places where Bohr's theory predicted that electrons would be. Yet Schrödinger's interpretation could not explain quantum tunnelling.

German physicist Max Born proposed a different interpretation that same year. Born stated that the square of the wave function does not represent the physical density of electron waves, but their probability density[8].

This is the probability of finding an electron in any particular state; that is, with any particular position, momentum, or energy, at any particular time.

The de Broglie model of the atom was now replaced with the idea that the electrons exist in a superpositional 'probability cloud'.

2. Quantum superpositions

During the double-slit experiment, it is the probability density that is 'waving', and the interference pattern is produced by the superposition of possible paths the electron could take.

Anything that can be described by the Schrödinger equation can be described as being in a superpositional state, where it exists in all possible quantum states at once.

A superposition is composed of all of the solutions to the Schrödinger equation and - since the Schrödinger equation is linear - there are often an infinite amount of solutions.

Linear equations are equations with the form a1x1 + a2x2 + ... + anxn = c, where c and a1...an are constants, and x1...xn vary. A linear equation with one variable, 3x = 9 for example, has one solution, x = 9/3 = 3. Liner equations with two or more variables have an infinite amount of solutions.

A Liner equations with two variables, y = 3x+3 for example, has possible solutions x=1, y=6, x=2, y=9, x=3, y=12 ... etc. and produces a straight line when plotted on a graph. With three variables, 2x + 3y - z = 9 for example, possible solutions include x=1, y=2, z=-4, x=2, y=2, z=1, x=2, y=1, z=-2 etc. and the equation produces a plane when plotted.

If, during the double-slit experiment, the position of the electron were measured, however, then a single result would be given with a probability of 100%. All other measurements would confirm this result, and an interference pattern would not form.

2.1 The collapse approach

Heisenberg interpreted the process of measurement as invoking a 'collapse' of the wave function, from a superpositional state into a single state, with a probability determined by Born's rule. This is known as the Copenhagen interpretation, or collapse approach to quantum mechanics[9].

The collapse approach suggests that the universe must be objectively indeterminate because you cannot predict what state a superposition will collapse into, you can only assign each possibility a probability.

This implies that you cannot know the future of the universe, even if you knew all of the physical laws and everything about its current state. Schrödinger and German-Swiss-American physicist Albert Einstein did not agree[10a].

3. The 1927 Solvay Conference on Physics

The search for the physical meaning behind these new equations was discussed at the 1927 Solvay Conference on Physics. This was attended by 29 scientists, including Erwin Schrödinger, Albert Einstein, Max Planck, Niels Bohr, Werner Heisenberg, Wolfgang Pauli, Louis de Broglie, Paul Dirac, Max Born, Marie Sklodowska-Curie, and Charles Thomson Rees Wilson, and Arthur Compton[7b].

In a joint paper delivered to the conference, Heisenberg and Born stated:

"we consider quantum mechanics to be a closed theory, whose fundamental physical and mathematical assumptions are no longer susceptible of any modification"[11].

Schrödinger and Einstein disagreed, and argued that quantum mechanics is a statistical approximation of an underlying deterministic theory[10b].

Part of the conference was filmed by American chemist Irving Langmuir, as shown below.

The 1927 Solvay Conference on Physics. 21 out of 29 attendees are shown, they are, in order of appearance: Erwin Schrödinger, Niels Bohr, Auguste Piccard, Werner Heisenberg, Paul Ehrenfest, Peter Debye, Wolfgang Pauli, Leon Brillouin, Hendrik Kramers, Paul Dirac, Max Born, Louis de Broglie, Irving Langmuir, Marie Sklodowska-Curie, William Lawrence Bragg, Arthur Compton, Owen Richardson, Hendrik Lorentz, Paul Langevin, Albert Einstein, and Max Planck. The 8 attendees not filmed were: Émile Henriot, Édouard Herzen, Théophile de Donder, Jules-Émile Verschaffelt, Ralph Howard Fowler, Martin Knudsen, Charles-Eugène Guye, and Charles Thomson Rees Wilson. Credit: Irving Langmuir via mikicorni.

4. References

  1. Schrödinger, E., 1926, 'An undulatory theory of the mechanics of atoms and molecules', Physical Review, 28, pp.1049-1070.

  2. Schrödinger, E., 1926, 'Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinem' ('On the Relationship of the Heisenberg-Born-Jordan Quantum Mechanics to mine'), Annalen der Physik, 384, pp.734-756.

  3. Beller, M., 2001, 'Quantum Dialogue: The Making of a Revolution', University of Chicago Press.

  4. Bohr, N., 1928, 'The Quantum Postulate and the Recent Development of Atomic Theory', Nature, 121, pp.580-590.

  5. Feynman, R. P., Leighton, R. B., and Sands, M., 1965, 'The Feynman Lectures on Physics, Volume III', Basic Books.

  6. Schrödinger, E., 1926, 'Quantisierung als Eigenwertproblem, vierte Mitteilung' ('Quantisation as a Problem of Proper Values, Part IV'), Annalen der Physik, 81, pp.109-139.

  7. (a, b) Bacciagaluppi, G. and Valentini, A., 2009 'Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference', Cambridge University Press.

  8. Born, M., 1926, 'Quantenmechanik der stoßvorgänge' ('On the quantum mechanics of collision processes'), Zeitschrift für Physik, 38, pp.803-827.

  9. Faye, J., 'Copenhagen Interpretation of Quantum Mechanics', Stanford Encyclopedia of Philosophy, last accessed 15-02-16.

  10. (a, b) Einstein, A., Podolsky, B. and Rosen, N., 1935, 'Can quantum-mechanical description of physical reality be considered complete?', Physical review, 47, pp.777-780.

  11. Born, M. and Heisenberg, W., 1927, 'Quantum mechanics', from 'Proceedings of the Fifth Solvay Congress' in Bacciagaluppi, G. and Valentini, A., 2009 'Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference', Cambridge University Press.

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