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Subject PHYSICAL CHEMISTRY
Paper No and Title 2, PHYSICAL CHEMISTRY-I
TOPIC QUANTUM CHEMISTRY
Sub-Topic (if any) THEORY OF ANGULAR MOMENTUM
Module No. CHE_P2_M15
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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TABLE OF CONTENTS
1. Learning outcomes 2. Angular momentum 3. Angular momentum operator 4. Relevance of angular momentum in quantum mechanics 5. Commutation relations 6. Summary
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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1. Learning Outcomes
After studying this module, you shall be able to
Understand the physical significance of angular momentum operator in quantum mechanics Learn about orbital and spin angular momentum Understand commutation properties of angular momentum operator
2. Angular momentum
Angular momentum in classical mechanics is defined by the equation:
푳⃗ = 풓⃗ × 풑⃗⃗ …(1)
Where r is particle’s position and p = mV is its linear momentum and L refers to angular momentum which is expressed as cross product of particle’s position and its linear momentum.
Since angular momentum, position and linear momentum are all vector quantities, these can be expressed in Cartesian coordinates as
̂ 푳⃗ = 푳⃗⃗⃗⃗풙 풊̂ + 푳⃗⃗⃗⃗풚 풋̂ + 푳⃗⃗⃗⃗풛 풌
풓⃗ = 풙⃗⃗ 풊̂ + 풚⃗⃗ 풋̂ + 풛⃗ 풌̂ …(2)
̂ 풑⃗⃗ = 풑⃗⃗⃗⃗풙 풊̂ + 풑⃗⃗⃗⃗풚 풋̂ + 풑⃗⃗⃗⃗풛 풌
which modifies equation (1) to
풊̂ 풋̂ 풌̂ …(3) ̂ 푳⃗⃗⃗⃗풙 풊̂ + 푳⃗⃗⃗⃗풚 풋̂ + 푳⃗⃗⃗⃗풛 풌 = ⌊ 풙 풚 풛 ⌋ 풑풙 풑풚 풑풛
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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The components of angular momentum along the respective axis are given by the following equation:
푳⃗⃗⃗⃗풙 = 풚⃗⃗ 풑⃗⃗⃗⃗풛 − 풛⃗ 풑⃗⃗⃗⃗풚
푳⃗⃗⃗⃗풚 = 풛⃗ 풑⃗⃗⃗⃗풙 − 풙⃗⃗ 풑⃗⃗⃗⃗풛 …(4)
푳⃗⃗⃗⃗풛 = 풙⃗⃗ 풑⃗⃗⃗⃗풚 − 풚⃗⃗ 풑⃗⃗⃗⃗풙
Angular momentum is always along the axis perpendicular to the plane of rotation. This means that if the body rotates in the y-z plane, then angular momentum is in x direction.
The magnitude of angular momentum is obtained by taking dot product of 푳⃗⃗ . 푳⃗⃗ which gives,
푳⃗⃗ . 푳⃗⃗ = |푳|ퟐ …(5)
2 2 2 2 퐿 = 퐿푥 + 퐿푦 + 퐿푧 …(6)
ퟐ ퟐ ퟐ ퟏ⁄ …(7) |푳| = (푳풙 + 푳풚 + 푳풛 ) ퟐ
Dot product of unit vectors 푖̂, 푗̂ 푎푛푑 푘̂ (mutually perpendicular to each other) 풊̂. 풊̂ = 풋.̂ 풋̂ = 풌̂. 풌̂ = ퟏ 풊̂. 풋̂ = 풋.̂ 풌̂ = 풌̂. 풊̂ = ퟎ 풋.̂ 풊̂ = 풌̂. 풋̂ = 풊̂. 풌̂ = ퟎ
The square of angular momentum is scalar
In quantum mechanics we encounter two types of angular momenta viz., (a) Orbital Angular
Momentum (L) due to circular motion of a particle through space and is analog of classical
angular momentum and (b) Spin Angular Momentum (S) which is an intrinsic property of a
particle unrelated to any sort of motion or with any classical mechanical significance. The Total CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
…(8)
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Angular Momentum (J) is the vector sum of orbital angular momentum and spin angular
momentum
푱 = 푳⃗ + 푺⃗
3. Angular momentum operator
The classical definition of angular momentum is carried over to quantum mechanics by replacing the coordinates and momenta in the classical equation (4) by their corresponding operators.
푳̂풙 = 풚̂풑̂풛 − 풛̂풑̂풚
…(9) 푳̂풚 = 풛̂풑̂풙 − 풙̂풑̂풛
푳̂풛 = 풙̂풑̂풚 − 풚̂풑̂풙
Where 풙̂, 풚̂ 풂풏풅 풛̂ are components of position operator 풓̂ and 풑풙, 풑풚 풂풏풅 풑풛 are components oflinear momentum operator풑̂, which are quantum mechanically expressed as:
Classical quantities Quantum mechanical operators 풙 풙̂ = 풎풖풕풊풑풍풚 풃풚 풙 풚 풚̂ = 풎풖풍풕풊풑풍풚 풃풚 풚 풛 풛̂ = 풎풖풍풕풊풑풍풚 풃풚 풛 풑 흏 풙 풑̂ = −풊ħ 풙 흏풙 풑 흏 풚 풑̂ = −풊ħ 풚 흏풚 풑 흏 풛 풑̂ = −풊ħ 풛 흏풛
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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풉 ħ = , 풉 = 푷풍풂풏풄풌′풔 풄풐풏풔풕풂풏풕 = ퟔ. ퟔퟐퟔ × ퟏퟎ−ퟑퟒ 풎ퟐ풌품/풔 ퟐ흅
The Cartesian components of angular momentum operator can now be expressed by substituting the position and linear momentum quantum mechanical operators in equation (9) which gives,
흏 흏 푳̂ = −풊ħ(풚 − 풛 ) 풙 흏풛 흏풚
흏 흏 푳̂ = −풊ħ(풛 − 풙 ) …(10) 풚 흏풙 흏풛
흏 흏 푳̂ = −풊ħ(풙 − 풚 ) 풛 흏풚 흏풙
4. Relevance of angular momentum in quantum mechanics
Quantum mechanics is used to study the properties of chemical systems. How is the concept of angular momentum relevant in quantum mechanics for our systems of interest? In order to calculate the properties of a chemical system under consideration quantum mechanically, the first step involves writing the Hamiltonian for the system. This Hamiltonian expression sums up the contributions from the kinetic energy of all nuclei, kinetic energy of all electrons, electron – nucleus electrostatic attraction, electron – electron electrostatic repulsion and nucleus – nucleus repulsion. This makes the complete Hamiltonian quite complicated with many terms. Even for the simplest molecular system: the hydrogen atom, the solution of the Hamiltonian turns out to be mathematically quite complicated. Hence, it becomes difficult to solve for the eigenfunctions of the Hamiltonian, which indeed determine the properties of the chemical system under consideration. So, a general approach is required to solve this problem. It is known that if two operators say A and B commute then the properties corresponding to these operators can be determined simultaneously and exactly. In other words, if operators퐴̂ and 퐵̂ commute, the eigenfunctions of operator 퐴̂ will also be the eigenfunctions of operator 퐵̂ which means that operators 퐴̂ and 퐵̂ share simultaneous eigenfunctions. [퐴̂, 퐵̂] = 0 푂푝푒푟푎푡표푟푠 푐표푚푚푢푡푒 CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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If one finds an operator that commutes with the Hamiltonian and is simpler to operate than Hamiltonian than one can solve for the eigenfunctions of the simpler operator first. This way, one obtains the eigenfunctions of the complex Hamiltonian. For this, what is required is a suitable operator that commutes with the Hamiltonian. One such operator might be the linear momentum of the system. The linear momentum of a system is conserved. However, the linear momentum operator (푝̂) does not commute with the Hamiltonian (퐻̂), [퐻̂, 푝̂] ≠ 0** (** Symbolic representation of linear angular momentum operator 푝̂)
Hamiltonian operator (퐻̂) is sum of kinetic energy operator (푇̂)and potential energy operator (푉̂).
푯̂ = 푻̂ + 푽̂ For molecular systems, the kinetic energy operator is square of momentum, 풑ퟐ 푲풊풏풆풕풊풄 풆풏풆풓품풚 = ퟐ풎
풑̂ퟐ 푻̂ = ퟐ풎
[푯̂, 풑̂] = [푻̂, 풑̂] + [푽̂, 풑̂]
ퟏ [푻̂, 풑̂] = [풑̂ퟐ, 풑̂] ퟐ풎 ̂ퟐ As per commutation rules, [풑̂, 풑 ] = ퟎ, which means that [푻̂, 풑̂] = ퟎ kinetic energy and linear momentum commute. The other commutation relation is of potential energy and momentum. Potential energy of a system is a function of its position. It is well known that position and momentum do not commute [풙̂, 풑̂] ≠ ퟎ, as a consequence of famous Heisenberg’s uncertainty principle which states that “ it is impossible to determine both position and momentum with equal exactness”. 풉 ∆풙. ∆풑 ≥ ퟒ흅 And thus potential energy does not commute with linear momentum. [푽̂, 풑̂] ≠ ퟎ. It is because of the potential energy term that the Hamiltonian operator does not commute with linear momentum operator.
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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The other choice of operator can be angular momentum. The total angular momentum of a system is conserved and it is known to commute with the Hamiltonian operator. Thus, the theory of angular momentum is relevant as the angular momentum operator commutes with the Hamiltonian which means that an eigenfunction of angular momentum operator must also be an eigenfunction of Hamiltonian operator. Before solving eigenfunctions and eigenvalues of angular momentum operator, we’ll first derive some important commutation relations between angular momentum components in the next section.
5. Commutation relations
The quantum mechanical operators for the components of orbital angular momentum are given
as,
흏 흏 푳̂ = 풚̂풑̂ − 풛̂풑̂ = −풊ħ (풚 − 풛 ) 풙 풛 풚 흏풛 흏풚
흏 흏 푳̂ = 풛̂풑̂ − 풙̂풑̂ = −풊ħ (풛 − 풙 ) 풚 풙 풛 흏풙 흏풛
흏 흏 푳̂ = 풙̂풑̂ − 풚̂풑̂ = −풊ħ (풙 − 풚 ) 풛 풚 풙 흏풚 흏풙
Since angular momentum is defined as a product of position operator and linear momentum
operator, we’ll first check the commutation of position and linear momentum components.
Case I:
[풙̂, 풑̂풙] 흏 휕 [풙̂, 풑̂] = [풙̂풑̂ − 풑̂풙̂] = [풙̂, 풊ħ ] (푝̂ = −푖ħ ) 풙 풙 풙 흏풙 푥 휕푥
흏풇(풙) 흏(풙풇(풙)) 흏풇(풙) 흏풇(풙) [풙̂, 풑̂]풇(풙) = −풊ħ [풙 + ] == −풊ħ [풙 − 풙 − 풇(풙)] = 풊ħ풇(풙) 풙 흏풙 흏풙 흏풙 흏풙 …(11) CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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So, 푥̂ and 푝̂푥do not commute that means that position and momentum of a particle in the same …(8) direction cannot be determined simultaneously with precision.
Case II:
[풙̂, 풑̂풚] 흏 휕 [풙̂, 풑̂] = [풙̂풑̂ − 풑̂풙̂] = [풙̂, 풊ħ ] (푝̂ = −푖ħ ) 풚 풚 풚 흏풚 푦 휕푦
흏풇(풙,풚) 흏(풙풇(풙,풚)) 흏풇(풙,풚) 흏풇(풙,풚) …(12) [풙̂, 풑̂]풇(풙, 풚) = −풊ħ [풙 + ] == −풊ħ [풙 − 풙 ] = ퟎ 풚 흏풚 흏풚 흏풚 흏풚
This means that momentum along y axis does commute with position operator in x direction.
Now, we’ll derive the commutation relations between the components of angular momentum operator.
Case study: [푳̂풙, 푳̂풚]
[푳̂풙, 푳̂풚]풇 = [푳풙푳풚 − 푳풚푳풙]풇 = 푳풙푳풚풇 − 푳풚푳풙풇 (a)
Solving 푳풙푳풚풇, 흏 흏 흏풇 흏풇 푳 푳 풇 = [−풊ħ (풚 − 풛 )] . [−풊ħ (풛 − 풙 )] 풙 풚 흏풛 흏풚 흏풙 흏풛
흏풇 흏ퟐ풇 흏ퟐ풇 흏ퟐ풇 흏ퟐ풇 푳 푳 풇 = −ħퟐ [풚 + 풚풛 − 풚풙 − 풛ퟐ − 풛풙 ] (b) 풙 풚 흏풙 흏풛흏풙 흏풛ퟐ 흏풚흏풙 흏풚흏풛
Solving 푳풚푳풙풇, 흏 흏 흏풇 흏풇 푳 푳 풇 = [−풊ħ (풛 − 풙 )] . [−풊ħ (풚 − 풛 )] 풚 풙 흏풙 흏풛 흏풛 흏풚
흏풇 흏ퟐ풇 흏ퟐ풇 흏풇 흏ퟐ풇 푳 푳 풇 = −ħퟐ [풛풚 − 풛ퟐ − 풙풚 + 풙 + 풙풛 ] (c) 풚 풙 흏풙흏풛 흏풚흏풙 흏풛ퟐ 흏풚 흏풛흏풚
Substiting 푳풙푳풚풇 from equation (b) and 푳풚푳풙풇 from equation (c) into equation (a) gives, 흏풇 흏풇 [푳̂, 푳̂]풇 = −ħퟐ (풚 − 풙 ) = 풊ħ푳̂ …(13) 풙 풚 흏풙 흏풚 풛
The cyclic permutations yield the fundamental commutation relations satisfied by the components of the angular momentum operator. CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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[푳̂풙, 푳̂풚] = 풊ħ푳̂풛 …(14) [푳̂풚, 푳̂풛] = 풊ħ푳̂풙
[푳̂풛, 푳̂풙] = 풊ħ푳̂풚
These three commutation relations are the foundation of the theory of angular momentum in quantum mechanics. Any three operators which satisfy the above commutation relations represent the components of an angular momentum operator.
The magnitude of angular momentum is given by ퟐ ퟐ ퟐ ퟐ 푳 = 푳풙 + 푳풚 + 푳풛 …(15)
Note: It is in fact L2 that commutes with the full Hamiltonian.
Now we will derive the commutation relation between L2and the components of angular momentum.
̂ퟐ Case study: [푳 , 푳̂풙] Substituting the value of 푳ퟐfrom equation (15) gives,
̂ퟐ ̂ퟐ ̂ퟐ ̂ퟐ [푳 , 푳̂풙] = [푳풙, 푳̂풙] + [푳풚, 푳̂풙] + [푳풛 , 푳̂풙]
̂ퟐ [푳 , 푳̂풙] = 푳̂풚[푳̂풚, 푳̂풙] + [푳̂풚, 푳̂풙]푳̂풚 + [푳̂풛, 푳̂풙]푳̂풛 + 푳̂풛[푳̂풛, 푳̂풙]
Taking into consideration the commutation relations between components 퐿푥, 퐿푦 푎푛푑 퐿푧 of angular momentum operator as given in equation (14), the above equation simplifies to:
̂ퟐ [푳 , 푳̂풙] = −풊ħ푳풚푳풛−풊ħ푳풛푳풚 + 풊ħ푳풚푳풛 + 풊ħ푳풛푳풚 = ퟎ ̂ퟐ ̂ [푳 , 푳풙] = ퟎ …(16)
Due to cyclic permutations, we also conclude that
̂ퟐ [푳 , 푳̂풙] = ퟎ …(17) CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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̂ퟐ [푳 , 푳̂풚] = ퟎ
̂ퟐ [푳 , 푳̂풛] = ퟎ
This means that L2commutes with each of its components. i.e., it is possible to specify an exact value of L2and any one component.
Interestingly, in classical mechanics, when angular momentum is conserved, each of its three components has a definite value. However, in quantum mechanics, when angular momentum is conserved, only its magnitude and one of its components can be specified.
The above commutation relations holds true for both the spin and orbital angular momentum respectively. From this module, we conclude that the three components of angular momentum operator cannot be specified or measured simultaneously. Only, the total angular momentum (magnitude) and any one component of angular momentum can share simultaneous eigenfunctions (simultaneously measurable), i.e., they commute. For instance, if L2commutes
with Lz component, then one cannot know what are the x- and y- angular momentum components with certainity. There always exists an uncertainity relation for every pair of operators that does not commute. The best that can be done in quantum mechanics is to specify the magnitude of an angular momentum operator along with one of its components.
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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6. Summary
1. Angular momentum is expressed as cross product of particle’s position and its linear momentum.
풊̂ 풋̂ 풌̂ ̂ 푳⃗ = 푳⃗⃗⃗⃗풙 풊̂ + 푳⃗⃗⃗⃗풚 풋̂ + 푳⃗⃗⃗⃗풛 풌 = ⌊ 풙 풚 풛 ⌋ 풑풙 풑풚 풑풛
2. Angular momentum is always along the axis perpendicular to the plane of rotation. 3. The quantum mechanical operators for the components of orbital angular momentum are
흏 흏 푳̂ = 풚̂풑̂ − 풛̂풑̂ = −풊ħ (풚 − 풛 ) 풙 풛 풚 흏풛 흏풚
흏 흏 푳̂ = 풛̂풑̂ − 풙̂풑̂ = −풊ħ (풛 − 풙 ) 풚 풙 풛 흏풙 흏풛
흏 흏 푳̂ = 풙̂풑̂ − 풚̂풑̂ = −풊ħ (풙 − 풚 ) 풛 풚 풙 흏풚 흏풙
4. The theory of angular momentum in quantum mechanics is relevant in context of that angular momentum operator commutes with the Hamiltonian. 5. The three components of angular momentum operator cannot be specified or measured simultaneously.
[푳̂풙, 푳̂풚] = 풊ħ푳̂풛
[푳̂풚푳̂풛] = 풊ħ푳̂풙
[푳̂풛푳̂풙] = 풊ħ푳̂풚
6. The total angular momentum (magnitude) and any one component of angular momentum can share simultaneous eigenfunctions i.e., they commute. ̂ퟐ [푳 , 푳̂풙] = ퟎ
̂ퟐ [푳 , 푳̂풚] = ퟎ
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum
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̂ퟐ [푳 , 푳̂풛] = ퟎ
CHEMISTRY PAPER:2, PHYSICAL CHEMISTRY-I MODULE:15,Theory of Angular Momentum