Quantum Mechanical Operators and Their Commutation …

1 CHAPTER 1 Quantum Mechanics I 29 Copyright Mandeep Dalal Quantum Mechanical Operators and Their Commutation RelationsAn operator may be simply defined as a mathematical procedure or instruction which is carried out over a function to yield another function. (Operator) . (Function) = (Another function) (67) The function used on the left-hand side of the equation (67) is called as the operand the function over which the operation is actually carried out. The operator alone has no significance but when operated over a certain mathematical description, these Operators can provide very detailed insights into those functions. Some of the simple illustrations of equation (67) are given below. i) Consider the differential operator d/dx whose operation has to be studied over the function y = x5.

2 Themathematical treatment is = 5= 5 4(68) The operation of d/dx on y means that the rate of change of function y the variable x. The expression x5 is the operand while the 5x4 is the final result of our differential operator. ii) Consider the integral operator (y) dx whose operation has to be studied over the function y = x5. Themathematical treatment is ( ) = 5( ) = 66(69) The operation of dx on y means that we can find the function whose derivative is x5. The expression x5 is the operand while the x6/6 is the final result of our integral operator. In a similar way, the multiplication of a function by a constant number, or taking the square and cube roots of any function are also the Operators which give some other function after operating them over the operand.

3 The symbol of the operator typically carries a cap over it ( ) which differentiates it from the function used in the whole procedure. LEGAL NOTICEThis document is an excerpt from the book entitled A Textbook of Physical Chemistry volume 1 by Mandeep Dalal , and is the intellectual property of the Author/Publisher. The content of this document is protected by international copyright law and is valid only for the personal preview of the user who has originally downloaded it from the publisher s website ( ). Any act of copying (including plagiarizing its language) or sharing this document will result in severe civil and criminal prosecution to the maximum extent possible under Textbook of Physical Chemistry volume I Copyright Mandeep Dalal Algebra of Operators Just like the normal algebra, the resultants like addition or the multiplication of Operators also follow certain rules; however, these rules are different from the typical algebra.

4 Some of the most important rules of operator algebra are given below. and subtraction of Operators : Let A and B as two different Operators ; f as the function that hasto be used as the operand. Then, the addition and subtraction of these two Operators must be carried out in the manner discussed below. ( + ) = + (70) and ( ) = (71) of Operators : If A and B as two different Operators ; and f as the function that has to be usedas operand. Then, the multiplication of these two Operators must be carried out in the manner discussed below. = (72) The interpretation of the above equation is that first we need to operate B on f, which would give us another function f , which in turn is further used as the operand for operator giving the final result f.

5 In other words, we can say that when multiplication of two or more Operators is used, we should follow from left to right. Moreover, the square or cube of a particular operator must be considered as double or triple multiplication of the operator itself; mathematically, it can be shown as given below. 2 = (73) At this point it also very important to discuss one of the most fundamental properties of operator multiplication, the Commutation relation or the Commutation rule. Consider two Operators , A and B which can be operated over the function f. = ; = ; = 3 (74) Now = ( 3)= 4=4 3 (75) And = ( 3)= (3 2)=3 3 (76) From equation (75) and (76), it the clear that in this case Buy the complete book with TOC navigation,high resolution images andno 1 Quantum Mechanics I 31 Copyright Mandeep Dalal (77) These Operators are said to be non-commutating with the commutator given below.

6 =4 3 3 3 (78) However, the two Operators are said to be commute if Their result is the same even after reverting Their order of application. Mathematically, it can be stated as given by equation (79). = (79) This is quite different from the normal algebra in which the product of two numbers is always the same irrespective of the order of multiplication ( = ). Summarizing the Commutation rule, it can be concluded that [ , ]= =0 (80) and [ , ]= 0 - (81) 3. Linear Operator: An operator is said to be a linear operator if its application on the sum of two functionsf and g gives the same result as the sum of its individual operations. Mathematically, it can be shown as given below. ( + )= + (82) For example, consider the differential operator A; with f and g as the functions which have to be used as the operand.

7 = ; =2 2; =3 2 (83) or ( + )= (2 2+3 2)= (5 2)=10 (84) or + = (2 2)+ (3 2)=4 +6 =10 (85) Hence, from equation (84) and equation (85), it is clear that the differential operator is clearly linear in nature. On the other hand, the square root operator is not linear as it does not give the same result when operated individually. Buy the complete book with TOC navigation,high resolution images andno A Textbook of Physical Chemistry volume I Copyright Mandeep Dalal Some Important Quantum Mechanical Operators One of the most basic and very popular Operators in Quantum mechanics is the Laplacian operator, typically symbolized as 2, and is given by the following expression. 2= 2 2+ 2 2+ 2 2 (86) The popular form of the Schrodinger equation can be written in terms of Laplacian operator as well.

8 2 2+ 2 2+ 2 2+8 2 2( ) =0 (87) or 2 +8 2 2( ) =0 (88) The Laplacian operator is pronounced as del squared . This operator is also a part of the mighty Hamiltonian operator which forms the basis for value evaluation for other Operators , as we have already discussed in the postulates of Quantum mechanics. The Hamiltonian operator is typically symbolized as and is given by the following expression. = 28 2 ( 2 2+ 2 2+ 2 2)+ (89) or = 28 2 2+ (90) The popular form of the Schrodinger equation is written in terms of the Hamiltonian operator as well. = (91) or [ 28 2 ( 2 2+ 2 2+ 2 2)+ ] = (92) or ( 28 2 2+ ) = (93) Furthermore, we know from the third postulate of Quantum mechanics that owing to the constant value of E (eigenvalue) the wave function can be labeled as eigenfunction.

9 Buy the complete book with TOC navigation,high resolution images andno 1 Quantum Mechanics I 33 Copyright Mandeep Dalal Therefore, the Schrodinger equation is also called as the eigen value equation . Simplifying this, we can say that ( )( )=( )( ) (94) The equation (94) is applicable to observables in the Quantum Mechanical world. For three dimensional systems, like the Hamiltonian, the operator can be obtained by summing the individual Operators along three different axes. For instance, some important three-dimensional Operators are: = 28 2 ( 2 2+ 2 2+ 2 2) (95) = 2 ( + + ) (96) The list of various important Quantum Mechanical Operators in one dimension, along with Their mode of operation is given below.

10 Table 2. Name and symbols of various important physical properties and Their corresponding Quantum Mechanical Operators . Physical property Operator Name Symbol Symbol Operation Position x Multiplication by x Position squared x2 2 Multiplication by x2 Position cubed x3 2 Multiplication by x3 Momentum px 2 Momentum squared px2 2 24 2 2 2 Kinetic energy = 22 28 2 2 2 Potential energy V(x) ( ) Multiplication by V(x) Total energy = + ( ) 28 2 2 2+ ( ) Buy the complete book with TOC navigation,high resolution images andno Textbook of Physical Chemistry volume I Copyright Mandeep Dalal Besides the record of different Operators presented in Table 2 , there still many Operators which are extremely important like angular momentum, parity, or the step-up step-down Operators .