Motion in Microcosm | Quantum Mechanics

Electric/electrostatic forces and magnetic forces of atoms and molecules

 

 

> On Moment

 

What is moment in physics?

From https://en.wikipedia.org/wiki/Moment_(physics).

In physics, moment is a combination of a physical quantity and a distance.

 

Examples:

  • Moment of force or torque: τ  =r * F.  (It is a first moment)
  • Moment of inertia: I = Σ *m* r2( I = Σ m r 2 ) {\displaystyle (I=\Sigma mr^{2})}. (It is analogous to mass in rotational motion. It is a measure of an object's resistance to changes in its rotation rate. It is a second moment of mass.)
  • Electric dipole moment : The electric dipole moment between a charge of –q and q separated by a distance of d is p = q * d. (It is a first moment)
  • Magnetic moment: (μ = Ι * Α)  (It is dipole moment measuring the strength and direction of a magnetic source).( p = q d ) {\displaystyle (\mathbf {p} =q\mathbf {d} )

 

 

> On Dipole Moment

 

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

Dipole moment may refer to:

 

 

 

> On Momentum

 

Please refer to "Motion in Macrocosm"

 

From https://en.wikipedia.org/wiki/Motion_(physics).

Momentum is a quantity which is used for measuring the motion of an object. An object's momentum is directly related to the object's mass and velocity.

 

Linear momentum

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

 

Angular or rotational momentum

From http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html#am.

The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity:

L = I * ω

The above relationship can be transformed according to what is mentioned at https://en.wikipedia.org/wiki/Angular_momentum as follows:

L = r * p

This is the cross product of the position vector r and the linear momentum p=m*v of the particle.

 

 

 

> The Structure of the Atom

 

>>  Introduction

 

Refererences from the Khan Academy section "Electronic Structure of Atoms" and subsection "Quantum numbers and orbitals":

Tutorial: The quantum mechanical model of the atom

Video tutorial: Quantum numbers 

 

 

Figure 1: Screen capture from Khan Academy video Quantum numbers.

 

 

As mentioned at the Khan Academy video Quantum numbers, 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, 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 the video at the link https://www.khanacademy.org/science/physics/quantum-physics/atoms-and-electrons/v/quantum-wavefunction, or that its charge is distributed in space (you may also refer to the previous video about the De Broglie wavelength).

 

In order to find the probability of this distribution in space we need the quantum numbers.

 

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 

azimuthal 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)

orbital magnetic quantum number

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 2 : 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 3: "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 notes that follow are based on 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).

 

 

 

>> Orbital angular momentum L 

 

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".

 

 

 

>> Orbital magnetic dipole moment μorb 

(Magnetic moment associated with the orbital angular momentum)

 

Classically, a charged particle that performs a rotational motion sets up the magnetic field of a magnetic dipole (Halliday D., Resnick R., Chapter 40 citing module 32-5). The magnetic dipole moment μ is related to the angular momentum L by:

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

 

The electron has an orbital magnetic moment μorb given by the same equation but it 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

 

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. Although scientists say that the electron does not spin, it has an intrinsic property that made scientists choose this term. 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 as follows:

 

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, +1/2 for "spin up" and -1/2 for "spin down".

 

 

 

>> 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.

(...)

 

 

 

>> 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.

 

 

>> 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.

 

 

 

> Electric charge moving in an electric and a magnetic field

 

Electric charge moving in an electric (electrostatic) field (velocity being perpendicular to its field lines)

F=qE

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfor.html#c2

 

 

Electric charge moving in a magnetic field (velocity being perpendicular to its field lines)

F=qvB

Lorentz force F acts as centripital force and the charge will do a circular motion:

F=Fc <=> qvB=mv2/R <=> R=mv/qB

 

The radius of the circular motion, termed gyroradius, Larmor radius, or cyclotron radius is given by the above equation.

 

The angular frequency of this circular motion is known as the gyrofrequency, or cyclotron frequency, and can be expressed as

ω=v/R=qB/m

 

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

Applications : Cyclotron, Mass Spectrometer

 

 

Electric charge moving in an electric and a magnetic field experiences the Lorentz force

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

F=qE+qvB

 

 

 

https://en.wikipedia.org/wiki/Lorentz_force#/media/File:LorentzLeiden2016.jpg

 

 

 

Electric/Electrostatic force on shared electron (pair) between two atoms – Covalent bond

 

Two atoms share an electron as they attract it with comparable electric (electrostatic) force – Electron attributed to both – Shared force/electrostatic interaction on electrons is termed a covalent bond

 

Hydrogen is the element with atomic number 1, which means it has one proton in its nucleus.

It has one electron in its outer shell. In order to form a stable outer shell that has two electrons, it needs another electron.

 

Fluorine is the element with the atomic number 9, which means it has nine protons in its nucleus.

It has two electrons in its first shell and seven in its outer shell. In order to form a stable outer shell that has eight electrons, it needs one electron.

 

A hydrogen atom will share its electron with a fluorine atom and in this way they both complete their outer shell. The electrons will be attracted by both nuclei via an electric force, an electrostatic force. How does that work out in space? Will the shared electrons be in the middle of the distance between the atomic centers? It depends on the strength of the electric force from each side.

 

The nucleus of fluorine with its 9 protons (positive charges) exerts a much more significant electric force on the electrons than the one-proton hydrogen nucleus. (We say that F is more electronegative than H).

 

As a result, the shared electrons will shift slightly towards the fluorine.

 

That means that in the HF molecule, the H side will have a proton on its own and a little further away there will be 9 protons and 10 electrons towards the F side. Therefore, the H side will be slightly positive and the F side with 10-9=1 electrons in surplus will have a negative charge.

 

 

Image location

 

Water or H20, constitutes a similar case.

 

Oxygen is the element with atomic number 8. It has 6 electrons in its outer cell and needs another two.

 

Two hydrogen atoms will share their electrons. The nucleus of oxygen (8 protons) exerts a much stronger electric force on the electrons than the one-proton hydrogen nucleus. As a result, the shared electrons will shift slightly towards the oxygen.

 

 

 

 

 

Electric/Electrostatic force between opposite charged ions - Ionic bond

 

Two atoms exchange an electron– Atoms are no longer electrically neutral but they become ions – Electrostatic interaction between ions is termed an ionic bond

 

Sodium is the element with atomic number 11, which means it has eleven protons in its nucleus.

It needs to discard one electron to form a stable outer shell of 8 electrons.

As mentioned, fluorine needs for the same reason one electron.

Sodium gives one electron and becomes a positively charged ion while fluorine gets one and becomes negatively charged. Opposites attract and we say that an ionic bond is formed between the two. The sodium chloride crystal is said to be ionically bonded.

 

 

 

 

Van des Waals forces: Electric/Electrostatic forces between dipoles (permanent or induced)

 

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

 

Electric/Electrostatic force between permanent dipoles (Keesom force)

We mentioned that the molecules of water and HF are dipoles. The (+) pole of one molecule attracts the (–) pole of the other. Generally, we expect that the hydrogen atom in all its interactions with strong electronegative atoms e.g. N, O, F will have its electron shifted away and will thereby generate a dipole.

 

Electric/Electrostatic force between permanent dipole and corresponding induced dipole (Debye force)

One example of an induction-interaction between permanent dipole and induced dipole is the interaction between HCl and Ar. In this system, Ar experiences a dipole as its electrons are attracted (to the H side of HCl) or repelled (from the Cl side) by HCl.[6][7] 

 

 

Electric/Electrostatic force between instantaneously induced dipoles generated by electron cloud fluctuations (London dispersion force)

 

From this wikipedia link:

 

"The third and dominant contribution is the dispersion or London force (fluctuating dipole-induced dipole), which arises due to the non-zero instantaneous dipole moments of all atoms and molecules. Such polarization can be induced either by a polar molecule or by the repulsion of negatively charged electron clouds in non-polar molecules. Thus, London interactions are caused by random fluctuations of electron density in an electron cloud. An atom with a large number of electrons will have a greater associated London force than an atom with fewer electrons. The dispersion (London) force is the most important component because all materials are polarizable, whereas Keesom and Debye forces require permanent dipoles. The London interaction is universal and is present in atom-atom interactions as well. For various reasons, London interactions (dispersion) have been considered relevant for interactions between macroscopic bodies in condensed systems.Hamaker developed the theory of van der Waals between macroscopic bodies in 1937 and showed that the additivity of these interactions renders them considerably more long-range.[4]"