The Schrödinger Equation

By Caitlin French

Introduction to Quantum Mechanics

Quantum Mechanics is the mathematical description of the structure and interactions of particles on atomic and subatomic scales.

It developed from Planck’s suggestion that energy is made up of individual units (quanta), Einstein’s photoelectric effect, de Broglie’s proposal of wave-particle duality of both energy and matter and Heisenberg’s uncertainty principle.

There has been much philosophical debate about the different interpretations of quantum mechanics, ranging from Bohr’s Copenhagen interpretation (criticised using the Schrödinger’s cat thought experiment) to the Many Worlds theory. Different formulations of quantum mechanics came about, including Heisenberg’s matrix mechanics and Feynman’s sum over histories or path integral approach.

Quantum mechanical processes are extremely important, having applications in computers and also in keeping us alive, as quantum tunneling allows nuclear fusion in the Sun.

Postulates of Quantum Mechanics

1. Particles can behave as waves (wave-particle duality), and their state is described by a wavefunction, Ψ(x). It can be solved using the Schrödinger equation.

2. The wavefunction is related to the probability density of the particle’s position. The area under the line gives the probability.

Probability density = P(x) = |Ψn|2
Probability = ∫|Ψn|2.dx = ∫P(x).dx = 1 (normalisation)

Normalisation is when we say the total probability of a particle being between two points is 1.
In other words, we are certain that the particle is somewhere in that region.

3. Observable classical quantities A correspond with quantum mechanical linear operators Â: 

ÂΨ = aΨ

4. Operators have hats on them, and are often differential operators, like d²/dx². Here, an operator (Â) acts on the wavefunction/eigenfunction (Ψ). This returns the same Ψ, but multiplied by a scalar constant/eigenvalue (a), which is the physical value for the variable.

For instance: 

Ĥ is the Hamiltonian operator. The eigenequation for energy is: ĤΨ(x) = EΨ, where E (the eigenvalue) is the observable energy of the wavefunction.

Energy

There are two main types of energy: kinetic and potential. We see this in a simple pendulum. The total energy in this closed system (ignoring air resistance) is constant, being the sum of (gravitational) potential energy and kinetic energy.

E = T(x) + V(x)

where:

E is total energy

T is kinetic energy

V is potential energy

Take a particle confined in a box. When confined, the particle may have some potential energy (for instance Coulomb energy, which is the attractive energy of the electron to a proton, for example) and some kinetic energy of confinement.

Heisenberg’s Uncertainty Principle

Heisenberg’s Uncertainty Principle states that we cannot simultaneously know the exact position (x) and exact momentum (p) of a particle. The smaller the uncertainty (or more the precise the measurement) of position, the higher the uncertainty (or less precise the measurement) of momentum, and vice versa.

When a particle is confined to a small space, there is a small uncertainty in position ∆x, meaning for the Heisenberg Uncertainty Principle to hold, ∆p (uncertainty in momentum) must increase. If momentum increases (∆p ≈ p), then velocity increases (as p = mv, where p is momentum, m is mass and v is velocity), so kinetic energy increases. This is the kinetic energy of confinement.

Schrödinger Equation

Time-independent Schrödinger Equation:

Note that we take the hat off the V for simplicity of writing and that, again, ћ = h/2π.

(-ћ²/2m)(d²Ψ/dx²) is the kinetic energy.

If we rearrange this in the form of an eigenequation, with an operator on the left and an eigenvalue on the right, we get:

The Infinite Potential Well

We confine our particle in a box of width a between 0 and a on the x-axis. The y-axis shows the potential energy (V). Fitting a wavefunction into a box is like making a standing wave on some string.

x < 0             V = infinite

0 < x < a       V = 0

x > a             V = infinite

Inside the well, V = 0, so Schrödinger’s equation becomes:

where k = √(2mE)/ћ

Differentiating the function twice needs to return the same function back again, multiplied by –k2. So we can solve this second order differential equation to get Ψ. (Note that this is the same differential equation as for simple harmonic motion.)

In order for this to be an eigenequation, we need solutions where the second derivative returns Ψ multiplied by a constant eigenvalue.

We then find that a solution which works is:

Ψ ∝ sin(kx) and Ψ ∝ cos(kx)
So the general solution is Ψ = Asin(kx) + Bcos(kx) where A and B are constants

Checking this:

Ψ = Asin(kx) 

dΨ/dx = Ak cos(kx)

d²Ψ/dx² = - Ak² sin(kx) = - k² Ψ ✓

Ψ = Bcos(kx) 

dΨ/dx = - Bk sin(kx)

d²Ψ/dx² = -Bk² cos(kx) = -k² Ψ ✓

In the same way you use boundary conditions to find the +c when you integrate, we now set boundary conditions to find our solution. 

From the diagram, Ψ = 0 at x = 0 and x = a. In the same way you would fit standing waves on a string:

Let’s first look at when x = 0

We can discard the cosine solution as cos(kx) becomes cos(0) = 1 (and we want it to be 0)

On the other hand, sin(0) = 0, which is correct. That is why the diagram shows a sine curve.

Now look at when x = a

To meet our boundary conditions, we must have sin(ka) = 0. This will only be true for certain values of k, where ka is a multiple of π (π radians = 180^o)

So kn = nπ/a for n=1,2,3…

This is quantisation! There are only certain, discrete values of kn (wavevectors) and E_n (energy levels) for which a standing wave will fit in the box.

Therefore, the width a must be an integer or half integer number of wavelengths. If we rearrange:

a = nπ/k

When n = 2 (Ψ2), 1 full wave fits in the box with wavelength λ, so a = λ and

kλ = 2π

So k = 2π/λ is the wave vector, kx is the angle and θ = kx = 2πx/λ is the phase.

Now the probability of finding the particle in a certain position will be:

P(x) = |Ψ(x)|2

which looks like:

Note that for a well centred at the origin with edges at –a/2 and a/2, the wavefunction of the particle is the cosine solution.

So what are the energies of a particle of mass m in an infinite potential well?

This shows that energy is also quantised and its magnitude is determined by a quantum number n.

We can calculate the energy levels by substituting in for a (the width of the box) and n (the number of the energy level).

If we now normalise the wavefunction (set the total probability of the particle being in the box equal to 1), we can find the constant A_n in our solution, which we will find is the square root of 2/a. This value for A_n is also the constant in front of the cosine solution, if we instead integrate the square of the cosine wavefunction between –a/2 and a/2.

So the quantum wavefunctions for a particle confined to a box of length a have the form:

Ψ_n(x) = √(2/a) sin(nπx/a) for n=1,2,3…

The Finite Potential Well

This is a finite well of depth V0 and width 2a. 

Classically, if the energy of the particle, E, was less than V0 (the height of the well), then the particle would not be able to escape. However, in quantum mechanics, the particle can escape even if E < V0.

Rearranging the Schrödinger equation:

In region B, V₀ = 0. So, as before:

with solutions Ψ_B ∝ sin(kx) and Ψ_B ∝ cos(kx)

In regions A and C, V = V₀. So:

In regions A and C, E < V₀, so these are classically forbidden. If the total energy E is less than the potential energy V₀, then the particle would have negative kinetic energy T.

The e means it is no longer oscillating but is exponential decay (as shown in the above diagrams in the regions outside the well).

When the solution inside the well is an even function like cos (for Ψ₁ or Ψ₃), we know the constant A = C. When the solution inside the well is an odd function like sin (for Ψ₂), A = -C

As the wavefunction extends to areas outside the well, there is a small probability that the particle could be found outside the well.

Quantum Tunneling

So what does this mean?

Imagine a ball on a hill. It does not have enough energy to roll over the top of the hill to the other side. This is the classical view. However, in quantum mechanics, the ball (if this was on a much smaller scale) would be able to tunnel through to the other side, “borrowing” energy before quickly paying it back. This is quantum tunneling!

References

A Cavendish Quantum Mechanics Primer, M. Warner & A.C.H. Cheung

The New Quantum Universe, Tony Hey and Patrick Walters

http://abyss.uoregon.edu/~js/ast123/lectures/lec06.html