The Chemistry Maths Book, Second Edition

496 Chapter 17Determinants

*The wave function of a system of bosons is totally symmetric, and the interchange of the coordinates of any

pair of identical bosons leaves the wave function unchanged.

functions. The electron is a member of the class of particles called fermions, particles

with half-integral spin (a particle with zero or integral spin is call a boson). The wave

function of a system of identical fermions is totally antisymmetric with respect to

the interchange of the coordinates (including spin) of the fermions; that is, the

interchange of the coordinates of any pair of fermions results in the change of sign

of the wave function.*This is just the property of an alternating function. Thus, if

the functionsf

1

,f

2

, =,f

n

in the determinant (17.54) represent the occupied states of

the nelectrons of a system, and ifx

1

, 1 x

2

,1=, 1 x

n

represent the nsets of coordinates

(including spin) of the electrons, then the functions are called spin-orbitals and the

determinant (17.54) is called a Slater determinant. Because the Slater determinant is

antisymmetric, the interchange (of the coordinates and spin) of any pair of electrons

results in a change of sign of the determinant. If two of the functions (spin-orbitals)

are the same then two rows of the determinant are equal and the determinant is zero.

This is an expression of the Pauli exclusion principle, that no two electrons can be in

the same state (spin-orbital).

17.8 Exercises

Section 17.1

1.Use determinants to solve the pair of equations

4 x 1 + 1 y 1 = 111

3 x 1 + 12 y 1 = 112

Evaluate:

2. 
   
   
   
   3. 
      
      
      
      4. 
         
      
      
      
      
   
   
   
   

Section 17.2

5. Use determinants to solve the equations

x 1 + 1 y 1 + 1 z 1 = 16

x 1 + 12 y 1 + 13 z 1 = 114

x 1 + 14 y 1 + 19 z 1 = 136

Evaluate the determinants by expansion along (i)the first row, (ii)the second column:

6. 
   
   
   
   7. 
      
      
      
      8. 
         
         
         
         9. 
            
         
         
         
         
      
      
      
      
   
   
   
   


7. (i)Find the cofactors of all the elements of

(ii)Confirm that the same value of the determinant is obtained by expansion along every

row and every column

123

201

111

−

−

032

201

260

132

012

004

−

−

111

110

112

−

−

235

012

341

cos sin

sin cos

nn

nn

θθ

θθ

01 −

− 23

20

03