##### Document Text Contents

Page 1

Hermitian operators

The operator P is defined as hermitian if its r,s matrix element has the property

Pr s ≡ ∫ψr* P ψsdτ = ∫(Pψr)* ψs dτ = ∫ψs (Pψr)* dτ = ∫[ψs* (Pψr*)]* dτ ≡ Psr*

In other words, the matrix elements related by the leading diagonal of P are complex

conjugates of each other.

Operators that are hermitian enjoy certain properties. The Hamiltonian

(energy) operator is hermitian, and so are the various angular momentum operators.

In order to show this, first recall that the Hamiltonian is composed of a kinetic energy

part which is essentially

m

p

2

2

and a set of potential energy terms which involve the

distance coordinates x, y etc. If we can prove that the various terms comprising the

Hamiltonian are hermitian then the whole Hamiltonian is hermitian.

The distance coordinate ‘operator’ x

The coordinate x is an operator insofar as it can ‘operate’ on a function f(x) to

produce another function x f(x), albeit just a multiple of the original one.

( ) ******** )( srrssrsrrs xdxxdxxdxxx ≡∫=∫=∫≡ ψψψψψψ

which shows that x is hermitian.

The momentum operator p

The operator for the linear momentum1 in the x direction is

xi

px ∂

∂

≡

. Let

us suppose that our space is one-dimensional, along the x axis so that

x∂

∂

can be

written

dx

d

. Its r,s matrix element is

[ ]

−=≡ ∫∫

∞

∞−

∞

∞−

∞

∞−

dx

dx

d

i

dx

dx

d

i

p rssrsrrsx

***)( ψψψψψψ

1 We have already seen that the operator for kinetic energy is

∂

∂

+

∂

∂

+

∂

∂

−=++

2

2

2

2

2

22222

2222 zyxmm

p

m

p

m

p zyx , i.e. this is the kinetic energy hamiltonian for a

particle in free space.

1

Page 8

Recall the application of symmetry in the MO treatment of pentalene

(JS symmetry course)

The π MOs of the pentalene molecule transform according to symmetries A1, A2, B1

and B2 in point group C2v, and we showed that they were

ψ (A1) = c1½(ϕ1 + ϕ3 + ϕ4 + ϕ6) + c2 √½(φ2 + φ5) + c7√½(φ7 + φ8)

ψ (A2) = c1’ ½(ϕ1 – ϕ3 + ϕ4 – ϕ6)

ψ (B1) = c1” ½(ϕ1 + ϕ3 – ϕ4 – ϕ6) + c2” √½(φ2 – φ5)

ψ (B2) = c1”’½(ϕ1 – ϕ3 – ϕ4 + ϕ6) + c7”’√½(φ7 – φ8)

A symmetry orbital like ψ(A2) does not interact with an orbital with different

symmetry like ψ(B2). In other words the matrix element <ψ (A2) H ψ (B2)> is zero,

as we’ll demonstrate. In recognising the interactions between atomic orbitals at the

Hückel level remember that all β parameters for non-nearest neighbours are zero.

<ψ (A2) H ψ (B2)>

= ∫[c1’ ½(ϕ1 – ϕ3 + ϕ4 – ϕ6) H c1”’ ½(ϕ1 – ϕ3 – ϕ4 + ϕ6) + c7’”√½(φ7 –

φ8)]dτ

= c1’c1”’ (1/4)[α + α – α – α + 0 + 0 . . . ] + (1/2√2) c1’ c2”’ [β + β – β – β + 0 .

+ 0 . . .]

= 0

So the H matrix elements between functions corresponding to different symmetries

(here A2 and B2) are zero. It is for this reason that we can treat the four symmetries

A1, B1, A2 and B2 separately.

If an operator P can be found which commutes with operator H (e.g. Hamiltonian)

then any pair of non-degenerate eigenfunctions ψ1 and ψ2 of P produce zero matrix

elements of H.

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Hermitian operators

The operator P is defined as hermitian if its r,s matrix element has the property

Pr s ≡ ∫ψr* P ψsdτ = ∫(Pψr)* ψs dτ = ∫ψs (Pψr)* dτ = ∫[ψs* (Pψr*)]* dτ ≡ Psr*

In other words, the matrix elements related by the leading diagonal of P are complex

conjugates of each other.

Operators that are hermitian enjoy certain properties. The Hamiltonian

(energy) operator is hermitian, and so are the various angular momentum operators.

In order to show this, first recall that the Hamiltonian is composed of a kinetic energy

part which is essentially

m

p

2

2

and a set of potential energy terms which involve the

distance coordinates x, y etc. If we can prove that the various terms comprising the

Hamiltonian are hermitian then the whole Hamiltonian is hermitian.

The distance coordinate ‘operator’ x

The coordinate x is an operator insofar as it can ‘operate’ on a function f(x) to

produce another function x f(x), albeit just a multiple of the original one.

( ) ******** )( srrssrsrrs xdxxdxxdxxx ≡∫=∫=∫≡ ψψψψψψ

which shows that x is hermitian.

The momentum operator p

The operator for the linear momentum1 in the x direction is

xi

px ∂

∂

≡

. Let

us suppose that our space is one-dimensional, along the x axis so that

x∂

∂

can be

written

dx

d

. Its r,s matrix element is

[ ]

−=≡ ∫∫

∞

∞−

∞

∞−

∞

∞−

dx

dx

d

i

dx

dx

d

i

p rssrsrrsx

***)( ψψψψψψ

1 We have already seen that the operator for kinetic energy is

∂

∂

+

∂

∂

+

∂

∂

−=++

2

2

2

2

2

22222

2222 zyxmm

p

m

p

m

p zyx , i.e. this is the kinetic energy hamiltonian for a

particle in free space.

1

Page 8

Recall the application of symmetry in the MO treatment of pentalene

(JS symmetry course)

The π MOs of the pentalene molecule transform according to symmetries A1, A2, B1

and B2 in point group C2v, and we showed that they were

ψ (A1) = c1½(ϕ1 + ϕ3 + ϕ4 + ϕ6) + c2 √½(φ2 + φ5) + c7√½(φ7 + φ8)

ψ (A2) = c1’ ½(ϕ1 – ϕ3 + ϕ4 – ϕ6)

ψ (B1) = c1” ½(ϕ1 + ϕ3 – ϕ4 – ϕ6) + c2” √½(φ2 – φ5)

ψ (B2) = c1”’½(ϕ1 – ϕ3 – ϕ4 + ϕ6) + c7”’√½(φ7 – φ8)

A symmetry orbital like ψ(A2) does not interact with an orbital with different

symmetry like ψ(B2). In other words the matrix element <ψ (A2) H ψ (B2)> is zero,

as we’ll demonstrate. In recognising the interactions between atomic orbitals at the

Hückel level remember that all β parameters for non-nearest neighbours are zero.

<ψ (A2) H ψ (B2)>

= ∫[c1’ ½(ϕ1 – ϕ3 + ϕ4 – ϕ6) H c1”’ ½(ϕ1 – ϕ3 – ϕ4 + ϕ6) + c7’”√½(φ7 –

φ8)]dτ

= c1’c1”’ (1/4)[α + α – α – α + 0 + 0 . . . ] + (1/2√2) c1’ c2”’ [β + β – β – β + 0 .

+ 0 . . .]

= 0

So the H matrix elements between functions corresponding to different symmetries

(here A2 and B2) are zero. It is for this reason that we can treat the four symmetries

A1, B1, A2 and B2 separately.

If an operator P can be found which commutes with operator H (e.g. Hamiltonian)

then any pair of non-degenerate eigenfunctions ψ1 and ψ2 of P produce zero matrix

elements of H.

8