Angular Momentum of Electron - De Broglie's Explanation, FAQs

Edited By Vishal kumar | Updated on Jul 02, 2025 04:45 PM IST

Introduction:
The rotating equivalent of linear momentum is angular momentum in physics. Because it is a preserved measure, the total angular momentum of a sealed system remains constant. It is a significant quantity in physics. Angular momentum of electron formula is mvr or nh/2. In this article we will study in detail about angular momentum of an electron, quantisation of angular momentum, property of an electron moving in a circular orbit and orbital angular momentum of an electron. Also, the quantised meaning in Tamil is அளவிடப்பட்டது.

Angular Momentum of Electron - De Broglie's Explanation, FAQs

Electron spin:

The spinning electron was successful in describing not only fine structure and the strange Zeeman Effect, but also other atomic phenomena.

The Bohr Sommerfeld quantum theory of elliptic orbits with relativity correction was in fair numerical agreement with the observed fine structure of hydrogen spectral lines. It however, suffered from two main drawbacks or downsides:

Firstly, Summerfield's relativistic explanation could not be applied for the spectral lines of atoms other than hydrogen. Alkali atoms' spectral lines, for example, are doublets with two near fine-structure components. The (single) optical electron in alkali atoms moves in a Bohr-like orbit with a large radius and low velocity. Therefore, the relativity effect would be too small to account for the large (compared with hydrogen) Fine-structure splitting observed in alkali lines.
Secondly, the simple quantum theory failed to explain the anomalous Zeeman Effect, that is, the splitting of atomic spectral lines into four, six or more components when the light source was placed in an external magnetic field.

In an effort to remove these two drawbacks of the theory, Goudsmit and Uhlenbeck proposed in 1925 that an electron must be looked upon as a changed sphere spinning about its own axis, having an intrinsic (built-in) angular momentum and consequently an intrinsic magnetic dipole moment. These are called 'spin angular momentum' S and 'spin magnetic dipole moment' M, respectively.

Let us write the magnitude S of the spin angular momentum of the electron in terms of a 'spin quantum number’s, as we do for the orbital angular momentum L in terms of orbital quantum number l. Thus S

$S=\sqrt{s(s+1)}\frac{h}{2\pi}$
The only value which the spin quantum number can be given is s = ½, which confirms the observed fine-structures doubling.

$S=\sqrt{\frac{1}{2}(\frac{1}{2}+1)}\frac{h}{2\pi}=\frac{\sqrt{3}}{2}\frac{h}{2\pi}$

The component of S along a magnetic field parallel to the z-direction is

$s_z=m_s\frac{h}{2\pi}$

Where ms is the 'spin magnetic quantum number' and takes (2s + 1) = 2 values which are +s and -s, that is

m_s= +1/2 and-1/2

The gyromagnetic ratio for electron spin μ_s/S is double the equivalent ratio μ_L/L=e/2m, according to experimental evidence.

The spinning electron was successful in describing not only fine structure and the strange Zeeman Effect, but also other atomic phenomena. Although electron spin was proposed as a postulate, Dirac established in 1928 that an electron must have an intrinsic rotational momentum and an intrinsic magnetic moment using relativistic quantum mechanics.

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Experimental evidence of electron spin:

• AH Compton proposed that “the electron is probably the ultimate magnetic particle” in Compton Scattering (1921).

• Stern-Gerlach Experiment (1922): A beam of silver atoms was passed through an inhomogeneous magnetic field, and two beams of space quantized components were seen.

• Uhlenbeck and Goudsmit (1925) demonstrated that there are two angular momentum states: intrinsic angular momentum of the electron and "SPIN" angular momentum.

• Pauli Exclusion Principle (1925): no more than two electrons per orbital, or no two electrons with identical quantum numbers. A new quantum number has been proposed.

Postulate 6: Under all conditions, all electronic wave functions must be ant symmetric.

graphical representation of angular and orbital angular momentum.

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Theoretical Justification:

Dirac (1928) developed relativistic quantum theory & derived electron spin angular momentum

Orbital Angular Momentum

L = orbital angular momentum

Formula of orbital angular momentum/Orbital angular momentum formula,

$L=h\sqrt{l(l+1)}$
l = orbital angular momentum quantum number

$l\leq n-1$
L_z = mℏ

m = 0, ±1, ±2… ±l

Spin Angular Momentum

S ≡ spin angular momentum

$S=h\sqrt{s(s+1)}=h\frac{\sqrt{3}}{2}$

s = spin angular momentum quantum number

s = ½

S = m_sℏ

m = ± 1/2

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Angular Momentum of Electron in Hydrogen Atom:

De Broglie's hypothesis can be used to compute the angular momentum of the hydrogen atom of an electron.

Consider an electron moving in a circular orbit of radius r.

According to the Broglie hypothesis, the associated wavelength of any particle is:

$\lambda =\frac{h}{M_v}....(1)$

From classical mechanics, the orbital angular momentum formula is:

L = Mvr …. (2) Also known as mvr formula

It is also noted that

2πr = mλ

$\lambda =\frac{2\pi r}{m}$

By inserting the value of λ in equation (1), we get:

$\frac{2\pi r}{m}=\frac{h}{M_v}...(3)$

From equation 1 we can get:

$M_v=\frac{L}{r}$

By substituting the Mv in equation (3) we get:

$\frac{2\pi r}{m}=\frac{h}{(\frac{l}{r})}=\frac{rh}{L}$

Now, solve this relation for angular momentum L:

$L=rh\times \frac{m}{2\pi r}$

or, $L=\frac{mh}{2\pi }$

or, $L=m.\frac{h}{2\pi}$

Here, $\frac{h}{2\pi}$ is designated as.

So, the angular momentum of the electron of Hydrogen atom is:

L = m.ℏ

Here, m = 1, 2, 3, 4 …so on

Also, the angular momentum is quantized in units of ħ.

In hydrogen atom,

The radius of n^th orbit is

r_n∝n²/Z

NCERT Physics Notes :

Where, Z is atomic number

So,

$n=(zr_n)^\frac{1}{2}$ ....(1)

Angular momentum of an electron in an orbit is given by,

$L=\frac{nh}{2\pi}.....(2)$

Put value of n from (1) in equation (2)

Angular momentum of electron in nth orbit, $L=\frac{(zr_n)^{\frac{1}{2}}h}{2\pi}$

L ∝ r^1/2

L∝ √r

Hence, angular momentum of electron in hydrogen atom is proportional tor.

We can now tell the angular momentum in different orbits, say, angular momentum of electron in 5th orbit is 25h.

The following is a summary of everything we've learned so far about the energy and dispersion of an electron in a hydrogen atom:

The energy grows as n grows, and it is solely dependent on n, the fundamental quantum number.
As n grows, the average distance between the electron and the nucleus increases.
As n increases, the number of nodes in the probability distribution grows.
As l is increased, the electron density concentrates along certain lines (or in planes).

Bohr’s quantisation condition states: In Bohr’s model, the angular momentum of electrons orbiting around the nucleus is quantized on a condition that the electrons can move only in orbits with an integral multiple of h/2 angular momentum.

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Frequently Asked Questions (FAQs)

1. What is the angular momentum of an electron in 3rd orbit?

The angular momentum of electron in 3rd orbit is 9h.

2. What is the orbital angular momentum of a 3d electron?

The orbital angular momentum of 3d electron is6(h2π).

3. How can you say that angular momentum of an electron is quantised?

In Bohr’s model, the angular momentum of electrons orbiting around the nucleus is quantized on a condition that the electrons can move only in orbits with an integral multiple of h/2 angular momentum.

4. What is the formula for orbital angular momentum?

Orbital angular momentum formula, L = ℏl(l+1).

5. What is the formula for spin angular momentum?

Spin angular momentum formula, S = ℏs(s+1).

6. What is the orbital angular momentum of p electron?

h/2

7. What is angular momentum, and how does it relate to electrons in atoms?

Angular momentum is a measure of rotational motion. For electrons in atoms, it describes their orbital motion around the nucleus. It's quantized, meaning it can only have specific, discrete values. This quantization is key to understanding electron behavior and atomic structure.

8. How did de Broglie's hypothesis contribute to our understanding of electron orbits?

De Broglie proposed that particles, including electrons, have wave-like properties. This wave-particle duality concept suggested that electron orbits could be described as standing waves around the nucleus. This idea helped explain why electrons occupy specific energy levels in atoms.

9. What is meant by the term "quantization" of angular momentum?

Quantization means that angular momentum can only have certain discrete values, not a continuous range. For electrons in atoms, this results in specific allowed energy levels and orbits, explaining the stability of atoms and their distinct spectral lines.

10. Why can't electrons have any arbitrary angular momentum in an atom?

Electrons can't have arbitrary angular momentum because of wave-particle duality. Their wave nature means they must form standing waves around the nucleus, which only occurs at specific orbits corresponding to whole number multiples of h/2π (where h is Planck's constant).

11. How does de Broglie's wavelength relate to electron orbits?

De Broglie's wavelength (λ = h/p, where p is momentum) must fit an integer number of times around the electron's orbit for a stable standing wave. This condition determines the allowed orbital radii and energy levels, explaining the quantization of electron states in atoms.

12. What is the significance of the principal quantum number in relation to angular momentum?

The principal quantum number (n) determines the electron's energy level and is directly related to its angular momentum. As n increases, both the energy and the angular momentum of the electron increase, corresponding to larger orbits farther from the nucleus.

13. How does the Bohr model of the atom relate to de Broglie's ideas?

The Bohr model proposed discrete electron orbits, which aligned with de Broglie's concept of electron waves. De Broglie's hypothesis provided a theoretical foundation for why only certain orbits were allowed, explaining the stability of atoms that the Bohr model described but couldn't fully explain.

14. What is the relationship between an electron's angular momentum and its magnetic moment?

An electron's angular momentum is directly proportional to its magnetic moment. This relationship is crucial in understanding atomic spectra and the behavior of atoms in magnetic fields. The quantization of angular momentum thus leads to quantized magnetic moments.

15. How does the uncertainty principle affect our understanding of electron orbits?

The uncertainty principle states that we cannot simultaneously know an electron's exact position and momentum. This limits our ability to describe electrons as following precise orbits. Instead, we use probability distributions (orbitals) to represent the likely locations of electrons around the nucleus.

16. What is the azimuthal quantum number, and how does it relate to angular momentum?

The azimuthal quantum number (l) determines the shape of an electron's orbital and is directly related to its angular momentum. It can have values from 0 to (n-1), where n is the principal quantum number. Each l value corresponds to a specific amount of angular momentum.

17. How do electron standing waves differ from classical waves?

Electron standing waves are probability waves, not physical waves like water or sound. They represent the probability of finding an electron in a particular location. Unlike classical waves, these quantum waves collapse to a single point when observed, following the principles of quantum mechanics.

18. Why do electrons not spiral into the nucleus despite constantly accelerating in their orbits?

According to quantum mechanics and de Broglie's hypothesis, electrons don't actually orbit like planets. Instead, they exist in quantum states with specific energy levels. The wave nature of electrons prevents them from existing between these levels, thus preventing them from spiraling into the nucleus.

19. How does the concept of angular momentum explain the stability of atoms?

The quantization of angular momentum restricts electrons to specific energy levels. This prevents them from losing energy continuously and collapsing into the nucleus. The discrete nature of these levels explains why atoms are stable and why they emit or absorb light at specific frequencies.

20. What is the role of Planck's constant in determining electron angular momentum?

Planck's constant (h) is fundamental in quantizing angular momentum. The allowed values of angular momentum are integer multiples of ħ (h/2π). This quantization is key to understanding the discrete energy levels and spectral lines of atoms.

21. How does the angular momentum of an electron change when it transitions between energy levels?

When an electron transitions between energy levels, its angular momentum changes by discrete amounts. These changes must comply with selection rules based on the conservation of angular momentum and energy. This results in the emission or absorption of photons with specific energies.

22. What is meant by the term "orbital angular momentum" in the context of electrons in atoms?

Orbital angular momentum refers to the angular momentum associated with the electron's motion around the nucleus. It's quantized and described by the azimuthal quantum number (l). This is distinct from the electron's intrinsic spin angular momentum.

23. How does the concept of standing waves explain the discrete energy levels in atoms?

Standing waves can only form when the electron's wavelength fits an integer number of times around its orbit. This condition is only met at specific orbital radii, corresponding to discrete energy levels. This explains why electrons can only exist at certain energy states within an atom.

24. What is the physical interpretation of the angular momentum quantum number?

The angular momentum quantum number (l) represents the magnitude of the electron's orbital angular momentum. It determines the shape of the electron's orbital and influences the electron's behavior in magnetic fields. Higher l values correspond to greater orbital angular momentum.

25. How does the angular momentum of an electron relate to its magnetic properties?

An electron's angular momentum generates a magnetic field, giving the electron a magnetic moment. This relationship is crucial in understanding atomic spectra, particularly the Zeeman effect, where spectral lines split in a magnetic field due to the interaction between the field and the electron's magnetic moment.

26. Why can't we use classical physics to accurately describe electron behavior in atoms?

Classical physics fails to explain atomic-scale phenomena because it doesn't account for the wave-like nature of particles. Quantum mechanics, which incorporates wave-particle duality, is necessary to accurately describe electron behavior, including quantized energy levels and probabilistic positions.

27. How does the angular momentum of an electron affect its energy in an atom?

The angular momentum of an electron is directly related to its energy. Higher angular momentum generally corresponds to higher energy levels. This relationship is key to understanding the structure of electron shells and the periodic table of elements.

28. What is the connection between de Broglie's wavelength and the Bohr radius?

The Bohr radius (the smallest allowed orbit in the Bohr model) can be derived using de Broglie's wavelength. It's the radius at which the electron's de Broglie wavelength fits exactly once around the circumference of the orbit, forming a standing wave. This connection highlights the importance of wave-particle duality in atomic structure.

29. How does the angular momentum of electrons contribute to the periodic table's structure?

The angular momentum of electrons determines their orbital shapes and energies. This, in turn, influences how electrons fill atomic orbitals, following the Aufbau principle, Hund's rule, and the Pauli exclusion principle. These filling patterns explain the periodic trends in element properties.

30. What is the significance of the magnetic quantum number in relation to angular momentum?

The magnetic quantum number (ml) specifies the orientation of the orbital angular momentum vector in space. It determines how the electron's angular momentum interacts with external magnetic fields, leading to phenomena like the Zeeman effect.

31. How does spin angular momentum differ from orbital angular momentum for electrons?

Spin angular momentum is an intrinsic property of electrons, while orbital angular momentum results from the electron's motion around the nucleus. Spin is quantized with only two possible values (up or down), whereas orbital angular momentum can have multiple quantized values depending on the orbital.

32. Why do electrons in s orbitals have zero orbital angular momentum?

Electrons in s orbitals (l = 0) have spherically symmetric probability distributions around the nucleus. This symmetry means there's no preferred direction of rotation, resulting in zero net orbital angular momentum. However, these electrons still possess spin angular momentum.

33. How does the angular momentum of electrons relate to chemical bonding?

The angular momentum of electrons influences the shape and orientation of atomic orbitals. This, in turn, affects how atoms can form chemical bonds. For example, the directional nature of p, d, and f orbitals (which have non-zero angular momentum) is crucial in understanding molecular geometry and bond types.

34. What is the relationship between an electron's angular momentum and its magnetic dipole moment?

An electron's magnetic dipole moment is directly proportional to its angular momentum. This relationship is described by the gyromagnetic ratio. Understanding this connection is crucial for interpreting atomic spectra and the behavior of atoms in magnetic fields.

35. How does the concept of angular momentum help explain the emission spectra of atoms?

The quantization of angular momentum leads to discrete energy levels in atoms. When electrons transition between these levels, they emit or absorb photons with specific energies. This results in the characteristic line spectra observed for different elements, each with a unique set of spectral lines.

36. What role does angular momentum play in the Stern-Gerlach experiment?

The Stern-Gerlach experiment demonstrated the quantization of angular momentum in atoms. It showed that the magnetic moment of atoms (related to angular momentum) could only have certain discrete orientations in a magnetic field, providing evidence for the quantum nature of atomic properties.

37. How does the angular momentum of electrons in multi-electron atoms differ from that in hydrogen?

In multi-electron atoms, electron-electron interactions complicate the angular momentum picture. While the basic principles remain the same, the presence of other electrons leads to phenomena like orbital hybridization and electron correlation, which affect the overall angular momentum states of the atom.

38. What is the connection between angular momentum and the fine structure in atomic spectra?

Fine structure in atomic spectra arises from the interaction between an electron's orbital angular momentum and its spin angular momentum. This spin-orbit coupling leads to slight energy level splits, resulting in closely spaced spectral lines that reveal more detailed information about atomic structure.

39. How does the uncertainty principle affect our ability to measure an electron's angular momentum?

The uncertainty principle limits our ability to simultaneously measure an electron's position and momentum with arbitrary precision. This affects angular momentum measurements because angular momentum is related to both position (radius vector) and momentum. We can only determine probabilities for angular momentum values in certain states.

40. Why is it impossible for an electron to have both zero energy and zero angular momentum simultaneously in an atom?

If an electron had both zero energy and zero angular momentum, it would be at rest at the nucleus. This is impossible due to the uncertainty principle, which prevents an electron from having a precisely defined position and momentum simultaneously. Additionally, the wave nature of the electron prevents it from being localized exactly at the nucleus.

41. How does the concept of angular momentum contribute to our understanding of atomic clocks?

Atomic clocks rely on transitions between specific angular momentum states of atoms. The precise energy differences between these states correspond to electromagnetic radiation at very specific frequencies. By measuring these frequencies, we can create extremely accurate time-keeping devices.

42. What is the relationship between an electron's angular momentum and its de Broglie wavelength?

An electron's angular momentum is quantized in units of ħ, while its de Broglie wavelength is given by λ = h/p. The connection between these concepts is that the allowed electron orbits are those where an integer number of de Broglie wavelengths fit around the circumference, satisfying both the quantization of angular momentum and the wave nature of the electron.

43. How does the angular momentum of electrons contribute to the magnetic properties of materials?

The angular momentum of electrons, both orbital and spin, gives rise to magnetic moments. In materials, the collective behavior of these magnetic moments determines macroscopic magnetic properties. Ferromagnetism, for example, arises from the alignment of electron spins, while orbital angular momentum contributes to phenomena like paramagnetism.

44. What is the significance of the total angular momentum quantum number in atomic physics?

The total angular momentum quantum number (J) combines the orbital (L) and spin (S) angular momenta of an electron. It's crucial for understanding the fine structure of atomic spectra and the behavior of atoms in magnetic fields. The allowed values of J determine the possible energy states and transition rules in complex atoms.

45. How does the angular momentum of electrons relate to the shape of electron orbitals?

The angular momentum of electrons, specifically the azimuthal quantum number (l), determines the shape of electron orbitals. s orbitals (l=0) are spherical, p orbitals (l=1) are dumbbell-shaped, d orbitals (l=2) have more complex shapes, and so on. These shapes reflect the angular distribution of electron probability density around the nucleus.

46. Why do transitions between certain angular momentum states in atoms produce circularly polarized light?

Transitions between certain angular momentum states can change the magnetic quantum number (ml) by ±1. This change in angular momentum is compensated by the emission of a photon with angular momentum, resulting in circularly polarized light. This phenomenon is crucial in understanding atomic transitions and light-matter interactions.

47. How does the concept of angular momentum help explain the Zeeman effect?

The Zeeman effect is the splitting of spectral lines in a magnetic field. It occurs because the external field interacts with the magnetic moment of the atom, which is directly related to its angular momentum. The different possible orientations of the angular momentum vector in the field correspond to slightly different energy levels, resulting in split spectral lines.

48. What is the connection between angular momentum and the selection rules for atomic transitions?

Selection rules for atomic transitions are based on the conservation of angular momentum. They dictate which transitions are allowed between different quantum states. For example, the rule Δl = ±1 for electric dipole transitions ensures that the change in orbital angular momentum is balanced by the angular momentum of the emitted or absorbed photon.

49. How does the angular momentum of electrons in atoms relate to the concept of electron shells?

Electron shells are groups of orbitals with the same principal quantum number (n). Within each shell, subshells are defined by the angular momentum quantum number (l). The allowed values of l within a shell determine the types of orbitals present (s, p, d, f), each with distinct angular momentum characteristics. This structure forms the basis of the periodic table.

50. Why is it important to consider both orbital and spin angular momentum when studying atomic spectra?

Both orbital and spin angular momentum contribute to the total angular momentum of an electron, which determines the fine structure of atomic spectra. The interaction between orbital and spin angular momenta (spin-orbit coupling) leads to energy level splitting, resulting in more complex spectral patterns that provide detailed information about atomic structure.

51. How does the angular momentum of electrons relate to the concept of electron configuration?

Electron configuration describes how electrons are distributed in an atom's orbitals. The Aufbau principle, Hund's rule, and the Pauli exclusion principle govern this distribution, all of which are based on the angular momentum properties of electrons. The order of filling orbitals (1s, 2s, 2p, etc.) reflects increasing energy and angular momentum.

52. What is the significance of the Larmor precession in relation to electron angular momentum?

Larmor precession describes the motion of the magnetic moment vector of an electron around an external magnetic field. This precession is directly related to the electron's angular momentum. The frequency of this precession (Larmor frequency) is proportional to the strength of the magnetic field and is important in phenomena like nuclear magnetic resonance.

53. How does the concept of angular momentum contribute to our understanding of the photoelectric effect?

While the photoelectric effect is primarily explained by the particle nature of light, angular momentum plays a role in the details of the process. The conservation of angular momentum in the interaction between photons and electrons influences the directionality of electron emission and helps explain certain aspects of the photoelectric effect in materials with complex band structures.

54. What is the relationship between an electron's angular momentum and its magnetic quantum number?

The magnetic quantum number (ml) specifies the projection of the electron's orbital angular momentum along a chosen axis (usually the z-axis in the presence of a magnetic field). It can take integer values from -l to +l, where l is the azimuthal quantum number. This quantization of angular momentum orientation is crucial in understanding atomic behavior in magnetic fields.

55. How does the angular momentum of electrons contribute to the stability of certain electron configurations?

Certain electron configurations, like those with completely filled or half-filled subshells, are particularly stable. This stability is partly due to the balanced arrangement of angular momenta in these configurations. For example, in a filled subshell, the orbital angular momenta of the electrons sum to zero, contributing to the overall stability of the atom or ion.