Motion in the Microcosm - Quantum Mechanics

Please refer to the page "Motion in Macrocosm | Classical Mechanics" for notions such as "angular momentum" which facilitate understanding of Quantum Mechanics.

Describing the dynamics of a quantum system, its total energy (potential and kinetic) - Hamiltonian vs Newtonian mechanics

In the Bohr model of the hydrogen atom, the electron is in orbit around the nucleus. However, similarly to the photon, which behaves like both a wave and a particle, the electron also has wave-like properties. This means that the electron is somehow "smudged out in space" as mentioned in this Khan Academy video, or that its charge is distributed in space. Classical mechanics can explain the behavior of a particle or a wave but not the combined behavior known as wave-particle duality. This is the objective of Quantum mechanics.

In the quantum mechanics version of the atom model, we don't know exactly where the electron is, but we can say with high probability that it is in an orbital. An orbital is the region of space where the electron is most likely to be found. It is also called an electronic shell or shell.

References: (a) Khan Academy video Quantum numbers, (b) Tutorial: The quantum mechanical model of the atom

To describe the dynamics of a quantum system we apply Hamiltonian mechanics as opposed to Newtonian mechanics.

The total energy of the system is represented by an operator termed a Hamiltonian, H, and as mentioned at https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics), by analogy with classical mechanics, the Hamiltonian in quantum mechanics is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system.

Also, please note the use of the coordinate system: "in Hamiltonian mechanics, a classical physical system is described by a set of canonical coordinates r = (q, p), where each component of the coordinate qi, pi is indexed to the frame of reference of the system".

Reference: Hyperphysics

Schrödinger equation and Wave Function

How do we describe the state of the system at time t?

The Hamiltonian generates the time evolution of quantum states, the Schrödinger equation as mentioned at this link:

In classical mechanics, by solving the equations of motion or Newton's laws you obtain the trajectory of a particle. In quantum mechanics, by solving the Schrondinger equation you obtain the wave function Ψ, which gives the probabilities for an electron to be find on a specific trajectory.

Describing the energy of the Hydrogen atom - solving the Schrödinger equation

We wish to describe the potential and kinetic energy of a quantum system. We will consider the example of the hydrogen atom system.

The potential energy is the result of position and configuration and in the case of the electron of the hydrogen atom this is due to the nucleus.

The potential energy is given by U= - e^2/ 4πεr.

As mentioned at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 the electron sees/senses a spherically symmetric potential.

Therefore, we use spherical polar coordinates to develop the Schrödinger equation.

As mentioned at the Wikipedia link https://en.wikipedia.org/wiki/Spherical_coordinate_system, and Figure 1, coordinates (r, θ, φ) as commonly used in physics (ISO convention) represent: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi).

For geographic coordinates we know that latitude specifies north-south position and longitude the east-west position. Colatitude is the complementary angle of a given latitude. The polar angle constitutes the colatitude. Azimuth is the angle of the object around the horizon (Figure 2).

Figure 1: (From Wikipedia) Spherical coordinates (r, θ, φ) as commonly used in physics (ISO convention).

Figure 2: (From Wikipedia - by TWCarlson) Horizontal coordinate system. Note that Azimuth is the angle of the object around the horizon.

In order to solve the Schrödinger equation, which is a partial differential equation, we can separate it into individual equations for each variable.

The solution is managed by separating the variables so that the wavefuction is represented by the product:

Ψ(r,θ,φ)= R(r)P(θ)F(φ)

The separation creates three equations, one for each variable. Solving the equation leads to the generation of a constant which is called a quantum number and the equation can have a solution only for specific values of each constant/quantum number. For instance, as mentioned at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c3:

The radial equation R(r) gives the principal quantum number n. A solution to the equation exists if and only if:

n= 1,2,3...

The colatitude equation P(θ) gives the orbital quantum number ℓ. A solution to the equation exists if and only if:

l= 0,1,2,3...n-1

The azimuthal equation F(φ) gives the magnetic quantum number m. A solution to the equation exists if and only if:

m= -ℓ, -ℓ+1, 0, ...ℓ-1, ℓ or 2ℓ+1 values.

In conclusion, the three spherical coordinates are associated to the three quantum numbers.

Figure  : "Wavefunctions of the electron of a hydrogen atom at different energies. The brightness at each point represents the probability of observing the electron at that point. (By PoorLeno - https://en.wikipedia.org/wiki/Wave_function).

Notes: For the sections below, in addition to the Wikipedia and Hyperphysics.phy-astr.gsu.edu links mentioned, the following textbook has been used as reference: Halliday D., Resnick R., Walker J., Priniciples of Physics, International Student Version, 10th Edition, Wiley, 05/2014 (Chapter 40 and Chapter 32, module 32-5).

Principal Quantum Number n and Energy Levels or Shells

https://en.wikipedia.org/wiki/Principal_quantum_number

The principal quantum number n represents the relative overall energy of each orbital. The energy level of each orbital increases as its distance from the nucleus increases. (Therefore, indirectly it represents the distance from the nucleus. The sets of orbitals with the same n value are often referred to as electron shells or energy levels."

"The minimum energy exchanged during any wave-matter interaction is the product of the wave frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. The difference between energy levels that have different n determine the emission spectrum of the element."

Orbital Angular Momentum L, Orbital Quantum Number ℓ, magnitude of L vector and Shape of Subshells

The Orbital Angular Momentum precesses around the z axis

https://en.wikipedia.org/wiki/Azimuthal_quantum_number

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydcol.html#c2

Similarly to an object that perfoms a rotational motion, each electron in an orbital is characterized by an orbital angular momentum L. This is the cross product r*p where r is the distance from the nucleus and p is the particle's linear momentum (v*m).

Let us examine the system which consists of a nucleus and an electron orbiting around the nucleus. According to the standard frame of reference, the electron orbits the nucleus. For our analysis, we will choose a different frame of reference, corresponding to that of an observer who is on the electron. For this alternative frame of reference, the electron is stationary and the nucleus orbits it. We refer to this as the rest frame of the electron. (Note: The observer sees the nucleus orbiting its location.)

The nucleus is a charged particle and as it moves in a closed loop, its movement is equivalent to a current I flowing in the loop as shown in the figure at the link "magnetic field in the electron frame". It is demonstrated at the figure, that this current I generates a magnetic field B. We are referring to the latter as an effective magnetic field in the (rest) frame of reference of the electron.

Due to this, the nucleus exerts a magnetic force on the electron (the electron "sees" or senses this). We can calculate the intensity (strength) of the magnetic field at the distance of the electron (r).  As this is due to a circular or orbital motion, an expression including the orbital angular momentum L of the electron is deduced for this magnetic field.

The magnetic field will exert a torque that will produce a change in the orbital angular momentum which is perpendicular to that angular momentum. This torque will tend to make the vector angular momentum L parallel to the magnetic field. As a result of the exerted torque, vector L will "tilt" and will be precessing around the direction of the magnetic field (z axis) with a certain angle as shown at this link http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/larmor.html#c1. This results in Larmor precession. We can calculate the precession angular velocity termed Larmor frequency which is associated with this precession movement.

Vector Model for Orbital Angular Momentum

As mentioned above and at the link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2, the orbital angular momentum for an atomic electron can be represented with a vector model where the angular momentum vector is shown to be precessing about a direction in space.

Note that the electron can have a defined magnitude for L but not a defined direction. We can find the magnitude L using the equation below, also written at the figure above, where L is associated to the orbital angular momentum quantum number, or orbital quantum number or azimuthal quantum number (ℓ):

L=SQRT (ℓ(ℓ+1)) * ħ           for ℓ = 0,1,2,... (n-1)

where

ℓ is the orbital quantum number

ħ is the reduced Planck constant equal to h/2π

n is the electron's principal quantum number

The orbital quantum number determines the magnitude of the orbital angular momentum and describes the shape of the orbital. The shape is denoted by the letters s, p, d and f is defined by values of the magnetic number ℓ, i.e. 0, 1, 2, or 3 respectively.

Note: sphere  (l = 0), polar (l = 1) or cloverleaf (l = 2)

"Their wavefunctions take the form of spherical harmonics, and so are described by Legendre polynomials.The various orbitals relating to different values of ℓ are sometimes called sub-shells, and (mainly for historical reasons) are referred to by letters, as follows:"

Azimuthal
number (ℓ)
Historical
Letter
Maximum
Electrons
Historical
Name
Shape
0 s 2 sharp spherical
1 p 6 principal three dumbbell-shaped polar-aligned orbitals; one lobe on each pole of the x, y, and z (+ and − axes); two electrons each lobe.
2 d 10 diffuse nine dumbbells and one doughnut (or “unique shape #1” see this picture of spherical harmonics, third row center)
3 f 14 fundamental “unique shape #2” (see this picture of spherical harmonics, bottom row center)

Figure 3: From Wikipedia By Unknown - Originally uploaded to :en by en:User:FlorianMarquardt at 18:33, 14 Oct 2002., CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=75046

The relationship between the magnitude of the orbital angular momentum and the orbital quantum number is commonly represented with a vector model as shown at the link

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c1 and described at this video https://youtu.be/m5KzP41GH9U?t=572

As mentioned at this link, there is only one way for a sphere (l=0) fo be oriented in space but orbitals that are polar (l=1) or cloverleaf (l=2) can point to different directions. Therefore a third quantum number is needed to describe the orientation of an orbital in space.

Magnetic quantum number ml, direction of vector L, number and polarity/orientation of orbitals

We can measure experimentally definite values of a component of L on a specific axis, usually z, as given by:

L= mℓ  *  ħ

where

mℓ  is the magnetic quantum number

ħ is the reduced Planck constant equal to h/2π

In other words although we cannot measure the vector L, we can determine its magnitude and only its projection along a specific axis (usually z). The z-component of the angular momentum is expressed using the magnetic quantum number mentioned above.

This z-component can take only certain values; in other words it is quantized as mentioned at this link http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2. Specifically, the orbital angular momentum is quantized to values of one unit of angular momentum apart. As mentioned above, its values are dependent on the magnetic quantum number ml. For instance for l=2 it can take the values m= -2, -1, 0, 1, 2.

The magnetic quantum number refers to the projection of the angular momentum in an arbitrarily-chosen direction, conventionally called the z direction or quantization axis. In other words, it is associated to the quantization of the z-component of angular momentum

Also, the magnetic quantum number refers loosely to the direction of the angular momentum vector.

With respect to the direction of the vector L, it is noted that using the above figure we can define a "semi-classical" angle θ given by:

cosθ = Lz/L

As mentioned in Wikipedia, "the magnetic quantum number only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of m are equivalent."

Also the magnetic quantum number (mℓ) was given this name to reflect the fact that it determines the energy shift of an atomic orbital due to an external magnetic field, a phenomenon known as the Zeeman effect.

Orbital magnetic dipole moment μorb

(Magnetic moment associated with the orbital angular momentum)

Let us consider an electron orbiting around a nucleus (e.g. a proton for the hydrogen atom). This movement of the electron in a closed loop corresponds to a current I flowing in the loop (please also refer to this link). It is known that a current-carrying wire or loop produces a magnetic field.

In order to express the magnetic strength of a magnet or an object that produces a magnetic field like an electric current loop we use the notion of the magnetic moment or more precisely the notion of the magnetic dipole moment in order to infer the equivalence to a magnetic dipole (a magnetic south pole and a magnetic north pole separated by a small distance).

As mentioned in Wikipedia (https://en.wikipedia.org/wiki/Magnetic_moment):

“The magnetic dipole moment of an object is readily defined in terms of the torque that object experiences in a given magnetic field. The same applied magnetic field creates larger torques on objects with larger magnetic moments. The strength (and direction) of this torque depends not only on the magnitude of the magnetic moment but also on its orientation relative to the direction of the magnetic field.”

The notion of magnetic moment represents the magnitude and the orientation of a magnetic source (e.g. magnet). A charged particle that performs a rotational motion sets up a magnetic field which constitutes a magnetic dipole and the configuration is equivalent to that of a closed loop run by current I. (Halliday D., Resnick R., Chapter 40 p.1107 citing module 32-5 p. 858 and Wikipedia reference).

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/orbmag.html

According to the Amperial loop model, the dipole moment of a loop which encloses surface A and is run by current I is:

m=I*A

Its direction is given by the right hand rule.

Also, as mentioned in Wikipedia, "the elementary magnetic dipole that makes up all magnets is a sufficiently small amperian loop of current I".

Following the derivation from this link, we integrate the orbital angular momentum L as mentioned below.

Reference for following notes:

https://en.wikipedia.org/wiki/Electron_magnetic_moment#Magnetic_moment_of_an_electron

"If the electron is visualized as a classical charged particle literally rotating about an axis with (orbital) angular momentum L, its magnetic dipole moment μ is given by

• {\displaystyle {\boldsymbol {\mu }}={\frac {-e}{2m_{\text{e}}}}\mathbf {L} ,}

where me is the electron rest mass.

It turns out that the classical result is off by a proportional factor for the spin magnetic moment. As a result, the classical result is corrected by multiplying it with a dimensionless correction factor g, known as the g-factor:

• {\displaystyle {\boldsymbol {\mu }}=g{\frac {-e}{2m_{\text{e}}}}\mathbf {L} .}

It is usual to express the orbital magnetic moment in terms of the reduced Planck constant ħ and the Bohr magneton μB:

• {\displaystyle {\boldsymbol {\mu }}=-g\mu _{\text{B}}{\frac {\mathbf {L} }{\hbar }}.}

The orbital magnetic moment (μorb) is quantized. We can find the quantized values by substituting L in the above equation. Similarly to what was mentioned above, we can calculate the value of μorb  but not its direction i.e. the vector. However, we can find the direction for a specific axis, usually the z axis.

Spin angular momentum S, Spin quantum number ms

An object rotates around an axis and if this axis passes through the body's center of mass then the object is said to rotate upon itself or spin. Experimental evidence linked to the Stern-Gerlach experiment and the Zeeman effect suggested that the electron posseses except for orbital angular momentum, an intrinsic angular momentum that could be related to a property of "spinning" like a ball in classical mechanics, although this could not be the case. The experiments suggested two different possible states and a value of 1/2.

We refer to the spin angular momentum S, or simply spin. Its magnitude is quantized:

S= [ SQRT(s(s+1)) ] * h          for s=1/2
where s is the spin quantum number

The electron is said to be a spin 1/2 particle. Same applies to protons and neutrons.

Similarly to what was mentioned previously, we can find the magnitude of the spin angular momentum but not its direction. However, we can do calculations for one axis, the z axis.

http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html#c3

"The spin (projection) quantum number (ms) describes the spin (intrinsic angular momentum) of the electron within that orbital, and gives the projection of the spin angular momentum S along the specified axis:"

Sz = ms ħ.

Sz=m* h

for ms= +/-s = +/-1/2

where ms is the spin magnetic number or spin quantum number, which can have only two values i.e. +1/2 for "spin up" and -1/2 for "spin down".

"In general, the values of ms range from −s to s, where s is the spin quantum number, an intrinsic property of particles [5]:"

• ms = −s, −s + 1, −s + 2, ..., s − 2, s − 1, s.

"An electron has spin number s = ½, consequently ms will be ±½, referring to "spin up" and "spin down" states".

Spin magnetic dipole moment μs

(Magnetic moment associated with the spin angular momentum)

In accordance to what was mentioned for the orbital angular momentum, a spin magnetic dipole moment is associated with the spin angular momentum.

Reference for following notes:

https://en.wikipedia.org/wiki/Electron_magnetic_moment#Magnetic_moment_of_an_electron

The spin magnetic moment is intrinsic for an electron.[2] It is

• {\displaystyle {\boldsymbol {\mu }}_{\text{s}}=-g_{\text{s}}\mu _{\text{B}}{\frac {\mathbf {S} }{\hbar }}.}

wher S is the electron spin angular momentum. The spin g-factor is approximately two: gs ≈ 2. The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body.

The z component of the electron magnetic moment is

• {\displaystyle ({\boldsymbol {\mu }}_{\text{s}})_{z}=-g_{\text{s}}\mu _{\text{B}}m_{\text{s}},}

where ms is the spin quantum number. Note that μ is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.

Total angular momentum J - Vector Model

If s is the particle's spin angular momentum and ℓ its orbital angular momentum, the total angular momentum j is:

j=s+ℓ

This is represented by the total angular momentum quantum number.

The above combination or addition can be represented by a vector model. Before describing this, let us consider some examples from classical mechanics.

Figure 3: From Wikipedia (selection) - By Maschen - Own work, CC0, included at https://en.wikipedia.org/wiki/Angular_momentum

The above figure (Figure 3) from Wikipedia shows on the left an object that rotates upon itself or "spins" and which has spin angular momentum S. On the right there is an object that performs a rotation around an origin at distance r (it is in orbit around an origin) and has orbital angular momentum L. We can calculate the total angulal momentum J by adding the two vectors L and S following the principle shown at the figure.

Figure 4: From Wikipedia - By Maschen - Own work, Public Domain, - https://commons.wikimedia.org/wiki/File:Angular_momentum_conservation.svg

The above figure (Figure 4) from Wikipedia also shows the principle of the addition of the vectors of the two angular momenta to calculate the total angular momentum.

We have described further above the vector model for orbital angular momentum L, where vector L is shown to be precessing around the z axis.

Similarly, we can consider that the spin angular momentum vector S is precessing around the same axis.

In the following figure which represents an electron with l=1 and s=1/2 or s=-1/2, the vectors of orbital angulal momentum L and spin angular momentum S are added in order to determine the total angular momentum J.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2

Similarly to the case of vector model for the orbital angular momentum (alone), the projection of the total angular momentum on the z axis is quantized to values differeing by one unit of angular momentum.

Total magnetic dipole moment

The total magnetic dipole moment resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum J by the equation:

• {\displaystyle {\boldsymbol {\mu }}_{J}=g_{J}\mu _{\text{B}}{\frac {\mathbf {J} }{\hbar }}.}

The g-factor gJ is known as the Landé g-factor, which can be related to gL and gS by quantum mechanics (cf. Landé g-factor for details).

Quantum numbers - Summary

Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. In the case of electrons, the quantum numbers can be defined as "the sets of numerical values which give acceptable solutions to the Schrödinger wave equation for the hydrogen atom".

Four quantum numbers can describe an electron in an atom completely.

Name Symbol Orbital meaning Range of values Value examples
principal quantum number n shell 1 ≤ n n = 1, 2, 3, …

orbital (angular momentum) quantum number

subshell (s orbital is listed as 0, p orbital as 1 etc.) 0 ≤ ℓ ≤ n − 1 for n = 3:
ℓ = 0, 1, 2 (s, p, d)

magnetic quantum number, (projection of angular momentum)

m energy shift (orientation of the subshell's shape) −ℓ ≤ mℓ ≤ ℓ for ℓ = 2:
mℓ = −2, −1, 0, 1, 2

spin magnetic number, spin quantum number,

spin projection quantum number

ms spin of the electron (−½ = "spin down", ½ = "spin up") −s ≤ ms ≤ s for an electron s = ½,
so ms = −½, ½

There is also the total angular momentum quantum number j and the projection of the total angular momentum along a specified axis mj.

Note:

If an electron is free, it has only its intrinic quantum numbers s and ms.
If an electron is "trapped in an atom", it has also the quantum numbers n,  and m.

Figure 4 : Single electron orbitals for hydrogen-like atoms with quantum numbers n=1,2,3 (blocks), ℓ (rows) and m (columns). Wikipedia - By Geek3 - Own work, Created with hydrogen 1.1, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=67681892 https://en.wikipedia.org/wiki/Quantum_number

The first three quantum numbers of the electron (n, l, m) and the Zeeman effect

Petit manège - Le spin des électrons https://youtu.be/tcpBwgN5hlE
Vidéo par le synchrotron “SOLEIL”

Vidéo courte sur le Synchrotron SOLEIL

(Transcript translation starting at t=54 - https://youtu.be/tcpBwgN5hlE?t=54):

In 1913 "Bohr's atomic model was describing correctly the hydrogen atom. The electron was rotating in circular orbits corresponding to authorized energy levels. By describing the orbits with positive integers, 1, 2, 3 etc. Niels Bohr was introducing the first quantum number of modern physics. As his model was struggling to describe multi-electron atoms, the German physicist Arnold Sommerfeld improved it in 1916 by providing electrons with two additional degrees of freedom: being able to rotate on elliptical orbits like the planets of the solar system as well as modify their trajectory in the presence of a magnetic field. Sommerfeld was thus adding two numbers: "l" the "orbital quantum number" and "m" the "magnetic quantum number". "

"Magnetic because the electrons behave like a small electrical circuit that is sensitive to external magnetic fields. This is the Zeeman effect, named after the Dutch physicist that discovered it twenty years ago when he studied the sodium spectrum. By approaching a magnet to the sodium lamp, the characteristic yellow line of sodium, is subdivided, proving that the electron experiences an electromagnetic force that modifies its energy levels."

In 1916, the electron thus lives with three quantum numbers."

Atomic angular momentum F, nuclear angular momentum I

The atomic angular momentum F [ref 1] is equal to the sum of the nuclear angular momentum (net nuclear spin) I and the electronic angular momentum J (total angular momentum of the electrons). J is the vector sum of L, the total orbital angular momentum of the electrons and S, the total spin angular momentum of the electrons.