Revision notes for quantum mechanics

QUANTUM 2

,Dirac Notation
The inner product : (T10) = 1 % T* d

The outer product : B =
1 tr) < $1


The expansion of a
generalT in terms of a discrete orthonormal basis is : V =

[G) =
En in)
The expansion of a general t in terms of a continuous
orthogonal basis
eg position
.
is : /T =
(8(b)() d

Consider the overlap of position eigenket and a general T (olT)
a : = ( (()(xkdx = ((x0)

orthogonality of the position eigenket implies thatok) & (x xo) = -




The modulus squared of the expansion Coefficient 1C(o) R is the probability per unit length of being
in position eigenstate (o)
given that one is in state /T. This probability per unit length is just the
.




/TGco) at position thatColTr T(xc0)
probability density eco ,
so =




Hence TN(x) =
<xlM]

Similarly Tr(p) =
< plTr
1
iPt
(xp) momentum written the position basis that (x(p) @p(x)
is
eigenstate in such 2ike
=
a =




Projection Operators and the Completeness Relation

Projection operator Ph : =
(n) (n) >
-



projects a ket onto state
eg .
PlM) =
(j) <j) [In) =

[C(j) < ilm) =
<(j)

Completeness relation decomposition of the identity : [Pn En> n
= = I


Identity operator I :
=
(d
Eg. Relationship between the momentum and position wave functions .




Momentum wavefunction ↑(p) :
=
< pli) >
-

insert the identity decomposition -
> ↑(p) = < plEIT)

Take the eigenkets in the decomposition to be position eigenkets : T(p) = (< p()(x)tr) da
-
M(x)
( (k) + badx (e)
*
*
↑(p) = =
(p(x) =
(p =




- -it
Vector Representation of States

Basis states can be represented by rectors : 17 = 17 = An =
1 = (10 ... 0) 12 = 201 ... 0 1n = 100 ... 1)


A
generalised states : ↑
= & Citi with N
= c = 1

, The vector representation of the outer product is : ↑ =r *s = Cd
*


*
d
*
. . . d
Cdi Cade ... Cdn*
Matrix Representation of Operators : :.....
And .*
* *
Cnd2 ...
Cndn
Consider a Hermitian operator A acting on a n-dimensional Hilbert space : A Xi) = Xi Zi it El ...
n3



Are (110
=I
The operator canbe written : Tr 4 . 7141/7 , ...
0



.
1 ...
0 0 The matrix A is the representation of the operator with respect
A = 0 % . . .
0 to the basis /1) .
:: '
..
00 In A Hermition operator is its The diagonal elements Air the
always diagonal in basis .
...

own are


eigenvalues of the operator .




Often A EI 12
we need to express an operator eigenbasis.
not in its own
Eg .
= 1
1
- Al Aiz

i j 2A12
We construct the matrix by matrix elements Aij Azi Azz
: = 2




The columns of the matrix of the coefficients of the images of the basis rectors
consist .




*
for a Hermitian operator : (ii) < ; < So that Aij Air This means that the matrix A
=
=
.
is the complex
conjugate
of its transpose .




Every operator can be written as a sum
of outer products :
= Aili) il =
Diagonalisation of Operators
Given the matrix representation A of operator A the eigenvalues and eigenvectors can
,
be found by
diagonalising the matrix
.




1) Set the determinant of the matrix to : A-1 I =
0



2) Solve the equations of X to find the roots which the
are
eigenvalues

3) Insert the eigenvalues individually into the eigenvalue equation to find the eigenvectors
(A -



XI)(j) (8) =




4) Normalise the eigenvectors

Commutation Relations for Angular Momentum Operators

In classical mechanics the particle is defined as [ Exp By the correspondance
angular momentum of a =
, .




their corresponding classical quantities the quantum operator
principle operators obey the same relations as
- -




for angular momentum is T ~x > : =




=
I ( - -
Hence
([ [y).
we can

=
derive commutator relations :

[yPz-EEy EPic <Pz] .
-c


=
(yPz zPc] [EPy zP] -[yPz xFz] (EPy
.
-




. .
+
.
x *z]
The the
only things that don't commute are a coordinate and its own momentum ,
2nd and 3rd terms are zero




([c [g] .
=

(yPz EPx] [zy xz]
.
+
,
=

y(z z]qx .
+ cc(z ,z]py =
ihkiy -y *) itIz =



- un
- it ih