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 total energy of a quantum system (potential and kinetic) - Schrödinger equation and Wave Function - Quantum Numbers

 

As mentioned at the Khan Academy video Quantum numbers (cf. also Tutorial: The quantum mechanical model of the atom), in the Bohr model of the hydrogen atom, the electron is in orbit around the nucleus. In the quantum mechanics version of the 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.

 

The dynamics of any quantum system are described by a quantum Hamiltonian, H, which is an operator corresponding to the total energy of the system. 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. The Schrödinger Hamiltonian generates the time evolution of quantum states, the Schrödinger equation.

 

In the Schrödinger equation we find the notion of the wave function Ψ, which suggests that an electron has wave-like properties. This is in accordance with the discovery of the photon behaving like a wave as well as particle. This may mean that the electron is somehow "smudged out in space" as mentioned in this Khan Academy video, or that its charge is distributed in space (you may also refer to the previous video about the De Broglie wavelength).

 

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

 

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

 

 

 

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 a 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

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

 

"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. 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 and the shape of subshells

 

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). The orbital angular momentum L of an electron, unlike that of a classic particle, is quantized, that is it has certain allowed values. We can find the quantized values by solving Schrödinger’s equation. This gives us the following equation 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)) * h          for ℓ = 0,1,2,... (n-1)

where

ℓ is the orbital quantum number

n is the electron's principal quantum number

 

We can find the value of L but we cannot calculate its direction, i.e. we cannot calculate the vector L. However, we can find the direction for a specific axis, usually the z axis.

 

Lz =mℓ * h          for m=-ℓ...-2, -1, 0, 1, 2,...ℓ

where

mℓ   is the orbital magnetic quantum number or magnetic quantum number 

 

It will become evident further down why the above quantum number was termed "magnetic".

 

In accordance with what was mentioned previously, the angular momentum vector L is a "special kind of vector" as its projection along a specific direction (e.g. usually z - the axis used for the polar coordinates in the analysis is chosen arbitrarily - reference to quantization axis)  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, as shown at Figure 4 which is found at the above link, it is quantized to values of one unit of angular momentum apart. Its values are dependent on the magnetic quantum number m. For instance for ℓ=2 it can take the values mℓ =-2, -1, 0, 1, 2.

 

Figure 4: Image from http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/vecmod.html#c2 

 

 

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.

 

 

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.

 

"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

 

 

Orbital magnetic dipole moment μorb 

(Magnetic moment associated with the orbital angular momentum)

 

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).

 

 

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

 

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. 

 

https://en.wikipedia.org/wiki/Quantum_number#Traditional_nomenclatures

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  (note for Zeeman effect)

 

As mentioned in Wikipedia, the magnetic quantum number (m) 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. However, at the time of the discovery of the effect, the spin was not known. It must be mentioned that the actual magnetic dipole moment of an electron in an atomic orbital is linked not only to the electron orbital angular momentum, but also to the electron spin angular momentum, expressed in the spin quantum number. The fact hat the latter was omitted was the cause for observations termed as the "anomalous Zeeman effect".

 

For the above reasons, we define a total angular momentum which is the sum of the orbital and spin angular momenta.

 

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.

 

 

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 a similar 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. See Landé g-factor for details.

 

 

 

 

 

>> Note on the quantum numbers of an electron

 

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.

 

 

 

 

Quantum numbers - Summary

 

From https://en.wikipedia.org/wiki/Quantum_number:

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 

(angular momentum)

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

 

 

Note that there is also:

  1. The total angular momentum quantum number:< >j = | ± s|,which gives the total angular momentum through the relation

     

    • J2 = ħ2 j (j + 1).
  2. The projection of the total angular momentum along a specified axis:< >mj = −j, −j + 1, −j + 2, ..., j − 2, j − 1, analogous to the above and satisfies

     

    • mj = m + ms and |m + ms| ≤ j.

 

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

 

The relation between the total angular momentum vector j and the total angular momentum quantum number j is given by the usual relation (see angular momentum quantum number)

  • {\displaystyle \Vert \mathbf {j} \Vert ={\sqrt {j\,(j+1)}}\,\hbar }

The vector's z-projection is given by

  • {\displaystyle j_{z}=m_{j}\,\hbar }

where mj is the secondary total angular momentum quantum number. It ranges from −j to +j in steps of one. This generates 2j + 1 different values of mj.

 

 

 

 

 

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

 

 

 

 

Figure 5: "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).

 

 

 

>> 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” 

 

Publication Facebook

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."

 

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