Subject PHYSICAL CHEMISTRY
Paper No and Title 2, PHYSICAL CHEMISTRY-I
TOPIC QUANTUM CHEMISTRY
Sub-Topic (if any) EIGEN VALUES OF ANGULAR MOMENTUM-LADDER OPERATORS Module No. CHE_P2_M16
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
TABLE OF CONTENTS
1. Learning outcomes 2. Angular momentum in quantum mechanics 3. Angular momentum eigen values
3.1 Ladder operators 3.2 Commutation relations
3.3 Eigen values of angular momentum component �� 3.4 Eigen values of total angular momentum �� 3.5 Relation between the eigen values of angular momentum component and total angular momentum
4. Summary
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
1. Learning Outcomes
After studying this module, you shall be able to
• Find the eigen values of angular momentum operator • Learn about the role of ladder operators
� • Understand the relation between the eigen values of �� and �
2. Angular momentum in quantum mechanics
A particle moving with linear momentum p at a position r relative to coordinate origin gives angular momentum or more specifically orbital angular momentum (L). Orbital angular momentum is related to circular motion of an object.
The Cartesian components of angular momentum operator are � � � = �� − �� = −�ħ � − � � � � �� ��
� � � = �� − �� = −�ħ � − � …(1) � � � �� ��
� � � = �� − �� = −�ħ � − � � � � �� ��
Since, the quantum mechanical coordinate operators as well as linear momentum operators are Hermitian, the components of angular momentum operator are also Hermitian.
Using the fundamental commutation property, it is known that the components of the angular momentum operator obey the following set of commutation relations:
��, �� = �ħ�� CHEMISTRY PAPER:…(2) 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
��, �� = �ħ��
��, �� = �ħ�� Although the components of angular momentum operator do not commute with each other, however, each of them is known to commute with the �! operator. The magnitude of angular momentum is given by
� � � � …(3) � = �� + �� + ��
� � , �� = � …(4) � � , �� = �
� � , �� = �
From this we conclude that, no two angular momentum components are compatible. Only, the total angular momentum �! and any one component of angular momentum can share simultaneous eigen states (simultaneously measurable).
The angular momentum operator commutes with the Hamiltonian which means that both can have simultaneous eigen functions. Since Hamiltonian for higher molecular systems becomes quite complex to solve, therefore, it becomes convenient to solve for eigen values and eigen functions of angular momentum operator. In this module, we will be deriving the eigen values of angular momentum operator.
3. Angular momentum eigen values
For quantum mechanical problems involving angular momentum, the key operators of interest are
! �! and � , as that they commute. The commutation relations of angular momentum operators are sufficient to deduce the eigen values of these angular momentum operators without referring to the coordinate system.
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
! Note: It is not necessary to consider angular momentum component �! in operation with � . One ! can consider either of the angular momentum components viz., �! or �! in operation with � .
! However, in this study we will consistently use the component �! in operation with � . In the next section, we shall obtain the eigen values for angular momentum using ladder operators.
3.1 Ladder Operators In order to obtain the eigen values of angular momentum operator, a set of angular momentum operators are combined as in the equation below:
�! = �� + ��� …(5)
�! = �� − ���
where �! refers to raising or step up operator and �! refers to lowering or step down operator.
Together, the operators �! and �! are called ladder operators. The naming convention for these ! operators will be clear once we deduce the eigen values of the �! and � . ! Since the total angular momentum � and angular momentum component �! commute, let Y be ! the common eigen function of �! and � .
��� = �� …(6)
��� = �� …(7)
where c and b are the eigen values in the respective expressions. Before obtaining the eigen values, we first look into some important ladder operator commutation relations.
3.2 Commutation relations
• [�!, ��]
�!, �� = [�� + ���, ��]
= ��, �� + � ��, �� = −�ħ�� − ħ�� = −ħ�+
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
�!, �� = −ħ�+ …(8) �!�� = ���! − ħ�+
• [�!, ��]
�!, �� = [�� − ���, ��]
= ��, �� − � ��, �� = −�ħ�� + ħ�� = ħ�−
�!, �� = ħ�− …(9) �!�� = ���! + ħ�−
� • [� , �!] � � � � , �! = � , �� + � � , �� = �
� � � , �� = � , �� = �
� …(10) � , �! = �
� • [� , �!] � � � � , �! = � , �� − � � , �� = �
� � � , �� = � , �� = �
� …(11) � , �! = �
3.3 Eigen values of angular momentum component ��
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
Considering the angular momentum component eigen value equation (7), we have
��� = ��
Operating on equation (7) with �! gives, �!�!� = �!�� …(12)
Using the results of equation (8) in equation (12) gives,
�!�! − ħ�! � = �!�� …(13) �! �!� = (� + ħ)�!�
Equation (13) tells that function �!� is the eigen function of angular momentum component
�! with the eigen value (b+ħ). This means that operating on the eigen function Y with the raising
operator �! converts Y into another eigen function of �! with eigen value higher by a factor of ħ in comparison to that of Y. Application of raising operator to equation (13) gives
! ! �! �! � = (� + 2ħ)�! �
Repeated application of raising operator gives,
! ! …(14) �! �! � = � + �ħ �! � � = 0,1,2 … ..
And likewise, the operation of lowering operator L- on equation (7) gives, ! ! …(15) �! �! � = � − �ħ �! � � = 0,1,2 … ..
The expressions obtained in equation (14) and (15) reveal that the application of raising and lowering operators respectively on the eigen function Y with eigen value b, generates a ladder of eigen values which differ by a factor of ħ at each successive step. It is for this reason the
operators �! and �! are called ladder operators as there successive application creates a “ladder”
of eigen states/eigen values of �! .
Note: The application of �! or �! destroys the normalization of eigen function Y.
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
3.4 Eigen values of total angular momentum L2
Considering the total angular momentum eigen value equation (6), we have
��� = �� It is known that the total angular momentum �! commutes with the angular momentum
component �! which means that the eigen functions of the angular momentum component �! are also the eigen functions of �!.
b+kħ
b+2ħ b+ħ b b-ħ b-2ħ
b-kħ
Figure 1: Ladder of eigenvalues of �!.
Operating on equation (6) with �! gives, ! …(16) �!� � = �!��
Using the results of equation (10) in equation (16) gives
! � �!� = �!��
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
Repeated application of raising operator gives,
! ! ! � �! � = �! �� …(17)
And likewise, the operation of lowering operator �! on equation (6) gives,
! ! ! …(18) � �! � = �! ��
The expressions obtained in equation (17) and (18) reveal that unlike the eigen values of angular
! momentum component �!, all the eigen functions of total angular momentum � have the same eigen value c.
3.5 Relation between the eigen values of angular momentum component �� and total angular momentum ��
The eigen values obtained for angular momentum component �! are of the form
�!�! = �!�! � = 0,1,2 … .. …(19) ! �ℎ��� �! = � ± �ħ ��� �! = �±� The eigen value obtained for total angular momentum �! follows the equation,
! ! …(20) � �! = ��! �ℎ��� �! = �±�
The set of eigen values of �! generated by ladder operators must be bounded, .i.e., the ladder cannot extend indefinitely.
Operating on equation (19) with �! gives, ! ! …(21) �!�! = �!�!�! = �! �! Subtracting equation (21) from equation (20) gives ! ! ! � �! − �!�! = ��! − �! �!
! ! ! …(22) �! + �! �! = (� − �! )�!
! ! The operator �! + �! is a non negative term (sum of squared variables) which means that b is limited by the condition CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
c ≥ b2 k …(23) 1 or c 2 ≥ bk
…(24) c ≥ bk ≥ − c k = 0,±1,±2....
Let us now consider bmax and bmin as the largest and smallest eigen values of �! respectively, which modifies equation (7) as
�!�!"# = �!"#�!"# …(25) � � = � � ! !"# !"# !"# …(26)
Operating on equation (25) with �! gives
�!�!�!"# = �!�!"#�!"# …(27) �! (�!�!"#) = (�!"# + ħ)�!�!"#
The eigen value obtained by application of raising operator is ħ times higher than bmax which is
contradicting as we defined bmax as the largest eigen value of �!. The only possibility to this
contradiction is that the function (�!�!"#) vanish, i.e.,
(�!�!!") = 0 …(28)
Operating on equation (28) with lowering operator gives, …(29) �!�!�!"# = 0
where the multiplication of ladder operators gives, �!�! = �� − ��� (�� + ���)
! ! = �! − ��!�! + ��!�! + �!
! ! = � − �! + �[�!, �!]
…(30) CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
! ! = � − �! − ħ�!
Substituting equation (30) into (29) gives,
! ! � − �! − ħ�! �!"# = 0
! � − �!"# − ħ�!"# �!"# = 0
� = �! + ħ� !"# !"# …(31)
On similar grounds, the lower limit is obtained as ! � = �!"# − ħ�!"# …(32)
Equating equation (31) and equation (32) gives, ! ! �!"# + ħ(�!"# + �!"#) − �!"# = 0 …(33)
Solving this quadratic equation for the unknown bmax gives two roots,
−�!"# …(34) �!"# = �!"# − ħ
Obviously bmax cannot be less than bmin and therefore the second root is rejected. Hence, the only possibility is
�!"# = −�!"# …(35)
From equation (19), we know that �! = � ± �ħ � = 01,2, … ..
…(36) �!"# − �!"# = �ħ
Substituting equation (35) in equation (36) gives,
2�!"# = �ħ �ħ � �!"# = 2 = �ħ � = 2 �ℎ��� � = 0, +1, +2, … … CHEMISTRY PAPER: 2, PHYSICAL… (37)CHEMISTRY -I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
1 3 �!"# = 0, 2 , 1, 2 , 2, … ..
And likewise, �ħ …(38) �!"# = − 2 = −�ħ
In general, by the virtue of ladder operators, there are an integral number of units between bmax
and bmin. And the eigen value c (independent of k) of the total angular momentum �! is obtained as ! 1 3 …(39) � = � � + 1 ħ � = 0, 2 , 1, 2 , 2, … ..
Putting the values of b and c in equations (6) and (7) respectively gives,
� � � � …(40) � � = � � + � ħ � � = �, � , �, � , �, … ..
��� = ��ħ� �� = −�, … … … … … … , +� …(41)
The range of eigen values of �! is limited by j and –j, which means a total of 2j+1 eigen states are
possible for angular momentum component �!.
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS
4. Summary
1. The eigen values of Lz are either integer or half integers, i.e., they differ by integral steps.
��� = �� �ħ � � = � = �ħ � = � ����� � = �, ±�, ±�, … …
��� = ��ħ� �� = −�, … … … … … … , +�
2. The eigen values of �! are bounded such that 2 c ≥ bk
3. The eigen value of �! is obtained as ��� = ��
� � � � � � = � � + � ħ � � = �, � , �, � , �, … ..
CHEMISTRY PAPER: 2, PHYSICAL CHEMISTRY-I MODULE:16, EIGEN VALUES OF ANGULAR MOMENTUM- LADDER OPERATORS