Translations
The previous discrete spectrum state vector formalism
can be generalized also to continuos cases, in practice, by
replacing
• summations with integrations
• Kronecker’s δ-function with Dirac’s δ-function.
A typical continuous case is the measurement of position:
• the operator x corresponding to the measurement of
the x-coordinate of the position is Hermitean,
• the eigenvalues {x0 } of x are real,
• the eigenvectors {|x0 i} form a complete basis.
So, we have
x|x0 i = x0 |x0 i
Z ∞
1 =
dx0 |x0 ihx0 |
−∞
Z ∞
|αi =
dx0 |x0 ihx0 |αi,
−∞
where |αi is an arbitrary state vector. The quantity
hx0 |αi is called a wave function and is usually written
down using the function notation
hx0 |αi = ψα (x0 ).
Obviously, looking at the expansion
Z ∞
|αi =
dx0 |x0 ihx0 |αi,
−∞
The effect of an infinitesimal translation on an arbitrary
state can be seen by expanding it using position
eigenstates:
Z
00
00
|αi −→ T (dx )|αi = T (dx ) d3 x0 |x0 ihx0 |αi
Z
=
d3 x0 |x0 + dx00 ihx0 |αi
Z
=
d3 x0 |x0 ihx0 − dx00 |αi,
because x0 is an ordinary integration variable.
To construct T (dx0 ) explicitely we need extra constraints:
1. it is natural to require that it preserves the
normalization (i.e. the conservation of probability) of
the state vectors:
hα|αi = hα|T † (dx0 )T (dx0 )|αi.
This is satisfied if T (dx0 )is unitary, i.e.
T † (dx0 )T (dx0 ) = 1.
2. we require that two consecutive translations are
equivalent to a single combined transformation:
T (dx0 )T (dx00 ) = T (dx0 + dx00 ).
3. the translation to the opposite direction is equivalent
to the inverse of the original translation:
T (−dx0 ) = T −1 (dx0 ).
4. we end up with the identity operator when dx0 → 0:
the quantity |ψα (x0 )|2 dx0 can be interpreted according to
lim T (dx0 ) = 1.
dx0 →0
the postulate 3 as the probability for the state being
0
0
0
0
localized in the neighborhood (x , x + dx ) of the point x .
It is easy to see that the operator
The position can be generalized to three dimension. We
0
denote by |x i the simultaneous eigenvector of the
T (dx0 ) = 1 − iK · dx0 ,
operators x, y and z, i.e.
where the components Kx , Ky and Kz of the vector K
|x0 i ≡ |x0 , y 0 , z 0 i
are Hermitean operators, satisfies all four conditions.
x|x0 i = x0 |x0 i, y|x0 i = y 0 |x0 i, z|x0 i = z 0 |x0 i.
Using the definition
T (dx0 )|x0 i = |x0 + dx0 i
The exsistence of the common eigenvector requires
commutativity of the corresponding operators:
we can show that
[xi , xj ] = 0.
Let us suppose that the state of the system is localized at
the point x0 . We consider an operation which transforms
this state to another state, this time localized at the point
x0 + dx0 , all other observables keeping their values. This
operation is called an infinitesimal translation. The
corresponding operator is denoted by T (dx0 ):
[x, T (dx0 )] = dx0 .
Substituting the explicit reprersentation
T (dx0 ) = 1 − iK · dx0
it is now easy to prove the commutation relation
[xi , Kj ] = iδij .
T (dx0 )|x0 i = |x0 + dx0 i.
The equations
The state vector on the right hand side is again an
eigenstate of the position operator. Quite obviously, the
vector |x0 i is not an eigenstate of the operator T (dx0 ).
T (dx0 ) = 1 − iK · dx0
T (dx0 )|x0 i = |x0 + dx0 i
can be considered as the definition of K.
One would expect the operator K to have something to
do with the momentum. It is, however, not quite the
momentum, because its dimension is 1/length. Writing
p = h̄K
we get an operator p, with dimension of momentum. We
take this as the definition of the momemtum. The
commutation relation
[xi , Kj ] = iδij
So, we have derived the canonical commutation relations
or fundamental commutation relations
[xi , xj ] = 0,
[pi , pj ] = 0,
[xi , pj ] = ih̄δij .
Recall, that the projection of the state |αi along the state
vector |x0 i was called the wave function and was denoted
like ψα (x0 ). Since the vectors |x0 i form a complete basis
the scalar product between the states |αi and |βi can be
written with the help of the wave functions as
Z
Z
hβ|αi = dx0 hβ|x0 ihx0 |αi = dx0 ψβ∗ (x0 )ψα (x0 ),
can now be written in a familiar form like
i.e. hβ|αi tells how much the wave functions overlap. If
|a0 i is an eigenstate of A we define the corresponding
eigenfunction ua0 (x0 ) like
[xi , pj ] = ih̄δij .
Finite translations
0
Consider translation of the distance ∆x along the x-axis:
T (∆x0 x̂)|x0 i = |x0 + ∆x0 x̂i.
We construct this translation combining infinitesimal
translations of distance ∆x0 /N letting N → ∞:
N
ipx ∆x0
T (∆x0 x̂) = lim 1 −
N →∞
N h̄
ipx ∆x0
= exp −
.
h̄
It is relatively easy to show that the translation operators
satisfy
[T (∆y 0 ŷ), T (∆x0 x̂)] = 0,
so it follows that
[py , px ] = 0.
Generally
[pi , pj ] = 0.
An arbitrary wave function ψα (x0 ) can be expanded using
eigenfunctions as
X
ca0 ua0 (x0 ).
ψα (x0 ) =
a0
The matrix element hβ|A|αi of an operator A can also be
expressed with the help of eigenfunctions like
Z
Z
hβ|A|αi =
dx0
dx00 hβ|x0 ihx0 |A|x00 ihx00 |αi
Z
Z
0
=
dx
dx00 ψβ∗ (x0 )hx0 |A|x00 iψα (x00 ).
To apply this formula we have to evaluate the matrix
elements hx0 |A|x00 i, which in general are functions of the
two variables x0 and x00 . When A depends only on the
position operator x,
A = f (x),
This commutation relation tells that it is possible to
construct a state vector which is a simultaneous
eigenvector of all components of the momentum operator,
i.e. there exists a vector
|p0 i ≡ |p0x , p0y , p0z i,
the calculations are much simpler:
Z
hβ|f (x)|αi = dx0 ψβ∗ (x0 )f (x0 )ψα (x0 ).
Note f (x) on the left hand side is an operator while f (x0 )
on the right hand side is an ordinary number.
so that
px |p0 i = p0x |p0 i,
ua0 (x0 ) = hx0 |a0 i.
py |p0 i = p0y |p0 i,
pz |p0 i = p0z |p0 i.
The effect of the translation T (dx0 ) on an eigenstate of
the momentum operator is
ip · dx0
ip0 · dx0
0
0
0
T (dx )|p i = 1 −
|p i = 1 −
|p0 i.
h̄
h̄
The state |p0 i is thus an eigenstate of T (dx0 ): a result,
which we could have predicted because
[p, T (dx0 )] = 0.
0
Note The eigenvalues of T (dx ) are complex because it is
not Hermitean.
Momentum operator p in position basis {|x0 i}
For simplicity we consider the one dimensional case.
According to the equation
Z
T (dx00 )|αi = T (dx00 ) d3 x0 |x0 ihx0 |αi
Z
=
d3 x0 |x0 + dx00 ihx0 |αi
Z
=
d3 x0 |x0 ihx0 − dx00 |αi
we can write
ip dx00
1−
|αi
h̄
Z
=
dx0 T (dx00 )|x0 ihx0 |αi
Z
dx0 |x0 ihx0 − dx00 |αi
Z
∂
= dx0 |x0 i hx0 |αi − dx00 0 hx0 |αi .
∂x
=
In the last step we have expanded hx0 − dx00 |αi as Taylor
series. Comparing both sides of the equation we see that
Z
∂
p|αi = dx0 |x0 i −ih̄ 0 hx0 |αi ,
∂x
or, taking scalar product with a position eigenstate on
both sides,
∂
hx0 |p|αi = −ih̄ 0 hx0 |αi.
∂x
In particular, if we choose |αi = |x0 i we get
hx0 |p|x00 i = −ih̄
∂
δ(x0 − x00 ).
∂x0
Taking scalar product with an arbitrary state vector |βi
on both sides of
Z
∂
p|αi = dx0 |x0 i −ih̄ 0 hx0 |αi
∂x
we get the important relation
Z
∂
hβ|p|αi = dx0 ψβ∗ (x0 ) −ih̄ 0 ψα (x0 ).
∂x
Just like the position eigenvalues also the momentum
eigenvalues p0 comprise a continuum. Analogically we can
define the momentum space wave function as
hp0 |αi = φα (p0 ).
We can move between the momentum and configuration
space representations with help of the relations
Z
ψα (x0 ) = hx0 |αi =
dp0 hx0 |p0 ihp0 |αi
Z
0
0
φα (p ) = hp |αi =
dx0 hp0 |x0 ihx0 |αi.
The transformation function hx0 |p0 i can be evaluated by
substituting a momentum eigenvector |p0 i for |αi into
hx0 |p|αi = −ih̄
∂
hx0 |αi.
∂x0
Then
∂
hx0 |p0 i.
∂x0
The solution of this differential equation is
0 0
ip x
hx0 |p0 i = C exp
,
h̄
hx0 |p|p0 i = p0 hx0 |p0 i = −ih̄
where the normalization factor C can be determined from
the identity
Z
hx0 |x00 i = dp0 hx0 |p0 ihp0 |x00 i.
Here the left hand side is simply δ(x0 − x00 ) and the
integration of the left hand side gives 2πh̄|C|2 δ(x0 − x00 ).
Thus the transformation function is
0 0
ip x
1
0 0
,
exp
hx |p i = √
h̄
2πh̄
and the relations
0
0
ψα (x )
= hx |αi =
φα (p0 )
= hp0 |αi
=
Z
dp0 hx0 |p0 ihp0 |αi
Z
dx0 hp0 |x0 ihx0 |αi.
can be written as familiar Fourier transforms
0 0
Z
ip x
1
√
dp0 exp
ψα (x0 ) =
φα (p0 )
h̄
2πh̄
Z
ip0 x0
1
√
dx0 exp −
φα (p0 ) =
ψα (x0 ).
h̄
2πh̄