Theoretical Background

1 The Schrödinger Equation

In quantum mechanics (QM), a particle is modeled by a wave function, which describes its probability to be in a certain point in space at some given time. In order to find this wave function, one has to solve the Schrödinger equation:
(1)  − i()/(t)Ψ(r⃗, t) = HΨ(r⃗, t)
Here, Ψ(r⃗, t) is the wave function and H is the Hamiltonian of the system, which is simply an expression for the total energy of the system, in which all the classical properties (such as the particle’s momentum) have been replaced by operators (such as derivatives).
For a single particle in some potential, the classical Hamiltonian is simply the kinetic and potential energies of the particle: H = Ek + Ep = (p2)/(2m) + V(r⃗, t).
In QM, this expression is transformed to: H =  − (2)/(2m)2 + V(r⃗, t).
In many cases (such as the ones we’ll discuss here), the Hamiltonian is time-independent and then the Schrödinger eq. can be solved separately in time and in space. By assuming that we can write Ψ(r⃗, t) = ψ(r⃗)⋅T(t), we can reduce eq. (1↑) to the time-independent Schrödinger equation Hψ(r⃗) = Eψ(r⃗), which for the single particle is:
(2)  − (2)/(2m)2ψ(r⃗) + V(r⃗)⋅ψ(r⃗) = Eψ(r⃗)
Here E is a scalar that describes the total energy of the particle.
Although eq. (2↑) can be difficult to solve, in the cases we’ll discuss here it can be solved exactly.

2 Energies in Infinite Potential Wells

In this section we’ll discuss the solutions of two cases, where a particle is confined inside an infinite potential well. For simplicity, we’ll consider a 1-dimensional space in most of our discussion, but we’ll present the solution for the 2-dimensional case as well.

2.1 Infinite Square Potential Well

A 1D infinite square potential well of length L, is described by the function:
V(x) =  0  0 < x < L       ∞  else
Plugging this in the 1D version of eq. (2↑), we get that for any x outside the range [0, L], ψ(x) must be equal to zero. For x ∈ [0, L], on the other hand, eq. (2↑) is reduced to:
 − (2)/(2m)(d2)/(dx2)ψ(x) = Eψ(x)
with the boundary conditions: ψ(0) = ψ(L) = 0.
Solving this equation, one finds that it has the infinite set of solutions:
ψn(x) = sin((nπ)/(L)x) for n ∈ {1, 2, ...}
with the corresponding energies:
E(n) = (2π2)/(2mL2)n2
This means that the particle that is bounded in the potential well, can have only a total energy value that fits the expression above.
A sketch of these wave functions and energy levels can be seen in figure 1↓.
figure inf_square.png
Figure 1 Wave functions and energy levels of a particle in a 1D infinite square potential well.
The figure is adapted from [1].

In the 2D case (where we have an L × L potential well), we get the energy spectrum:
E(nx, ny) = (2π2)/(2mL2)(nx2 + ny2)  for nx, ny ∈ {1, 2, ...}

2.2 Infinite Parabolic Potential Well

A 1D infinite parabolic potential well is described by the function:
V(x) = (1)/(2)mω2x2
where m is the mass of the particle that is inside the potential well, ω is the particle’s angular frequency of vibration and x is the particle’s position relatively to the equilibrium (x = 0).
This is the potential of a harmonic oscillator.
The 1D Schrödinger’s equation for this potential becomes:
 − (2)/(2m)(d2)/(dx2)ψ(x) + (1)/(2)mω2x2ψ(x) = Eψ(x)
Solving this equation, we get the following infinite set of allowed energy levels, that the particle can have:
E(n) = ℏω((1)/(2) + n) for n ∈ {0, 1, 2, ...}
A sketch of the energy levels of the parabolic potential can be seen in figure 2↓.
figure inf_parabolic.png
Figure 2 Energy levels of a particle in a 1D infinite parabolic potential well.
The figure is adapted from [1].

In the 2D case, where the potential is given by V(x, y) = (1)/(2)mω2(x2 + y2), we get the energy spectrum:
E(nx, ny) = ℏω(1 + nx + ny) for nx, ny ∈ {0, 1, 2, ...}

3 Quantum Numbers and Degeneracy

In the cases above, the solutions to the Schrödinger equation are characterized by one or more quantum numbers, which can be equal to discrete integer values. Since different solutions (wave functions) describe different states of a particle, this means that each set of different quantum numbers can describe a different particle. For example, in the case of the 2D infinite parabolic potential, two different states of a particle (wave functions) are: ψ(nx = 1, ny = 0) and ψ(nx = 1, ny = 1).
In general, different states may have different energies, but this is not always the case. Sometimes we have different states that have the same energy. Then, we say that there is a degeneracy in the energy level.
In the 1D potentials we described above, there is no degeneracy in the energies, because there is only one quantum number. But in the 2D case there are degeneracies. For example, in the case of the 2D infinite parabolic potential, two degenerate states are ψ(nx = 2, ny = 0) and ψ(nx = 1, ny = 1). They are degenerate, because they have the same energy: E(2, 0) = E(1, 1) = 3ℏω.

4 Spin

An additional quantum number, that each particle possess, is the spin quantum number.
Its existence was postulated after experiments that were trying to verify the quantum nature of the angular momentum of some particles, resulted in measurements that did not agree with the theory available at that time. Therefore, it was suggested that in addition to its angular momentum, each particle holds some inherent angular momentum (which we’ll call spin), that may be thought of as the particle’s rotation around its “poles”, just as the earth rotates around its poles.
This spin, like the angular momentum of a particle, can be described by two quantum numbers: s and ms, where ms can be any value of the set ms ∈ { − s,  − s + 1, ..., s − 1, s} and s is an inherent property of the particle that doesn’t change (like its charge). But, unlike angular momentum, s can be any half integer. The electron, for example, has a spin s = (1)/(2), so it can either be in the spin state ms =  − (1)/(2) or ms = (1)/(2).
Moreover, the spin has an effect on the energy levels only in the presence of a magnetic field. Then the additional element  − μsB⃗ is added to the Hamiltonian, where B⃗ is the magnetic field and μs is the spin magnetic moment. In the case there is no magnetic field, the spin does not change the energy levels, but it adds some additional degeneracy, because it adds an additional quantum number.
For example, in the case of an electron in a 1D parabolic infinite potential, where before there was no degeneracy, we can see that now each energy level has two fold degeneracy. This is because, for each n ∈ {0, 1, 2, ...} there are two different choices of the quantum number ms that will give two different wave functions. For instance, the states ψ(n = 1, ms = (1)/(2)) and ψ(n = 1, ms =  − (1)/(2)) are two different states that have the same energy E(n = 1) = (3)/(2)ω, and so they are degenerate.
Despite the fact that it is useful to think about this property as the spinning motion of a particle around its “poles”, it is also very wrong. The spin is a purely quantum mechanical phenomena. There is no classical equivalent to this phenomena, so it is best to think about it as an inherent property of the particle.
One may find additional information on the subjects of angular momentum in QM and on the spin, in most books about QM and in particular, in [1], pp 1266–1272 .

5 Pauli’s Exclusion Principle

The Pauli exclusion principle states:
No two electrons can ever be in the same quantum state; therefore, no two electrons in the same atom can have the same set of quantum numbers. [1], p 1273.
This principle has a more general form that states that no two fermions can be in the same quantum state. Bosons, on the other hand, can be in the same quantum state and in fact, they tend to “prefer” to be in the same quantum state.
Fermions and bosons are simply two classes to which any particle belong. A fermion is any particle that has a half integer spin (such as the electron, which has a spin s = (1)/(2)). A boson is any particle that has an integer spin (such as the Z boson, which is an elementary particle that has a spin s = 1).


[1] R.A. Serway, J. John W. Jewett, V. Peroomian, “Physics for Scientists and Engineers”. 2010. URL