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European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
35
From the Quantum Mechanical State Space to the Position
and Momentum Spaces through a Simple Relation
Spiros Konstantogiannis
4 Antigonis Street, Nikaia 18454
Athens, Greece
spiroskonstantogiannis@gmail.com
(Received 20.01.2020, Accepted 23.08.2020)
Abstract
In an attempt to address the need for an alternative presentation of the quantum mechanical position
and momentum spaces, we provide a presentation that is more constructive and less calculative than
those found in literature. Our approach is based on a simple, intuitively understood relation that
expresses the physical equivalence of the quantum mechanical state space to the position and
momentum spaces. With this work, we hope to offer a perspective complementary to those found in
standard quantum mechanics textbooks.
Keywords: Position space, momentum space, state space, position operator, momentum operator.
INTRODUCTION
In 1926, Schrödinger developed wave mechanics, a formulation or representation of
quantum mechanics which is based on the idea that the quantum systems are
described by wave functions satisfying a wave equation, which is known as the
Schrödinger equation (Aspect & Villain, 2017; Gieres, 2000). Schrödinger also
demonstrated the physical equivalence of wave mechanics to matrix mechanics, the
other known at that time formulation of quantum mechanics, that Heisenberg, Born,
and Jordan had developed (Aspect & Villain, 2017; Gieres, 2000). In the following
years until 1931, Dirac, Jordan, and von Neumann developed a representation-free or
invariant formalism of quantum mechanics, according to which each quantum system
is associated with a separable, infinite-dimensional, complex Hilbert space, which is
known as state space (Gieres, 2000; Van Hove, 1958). The elements of the state space
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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are Dirac kets of finite norm representing possible bound states of the examined
system (Gieres, 2000).
In the framework of the Hilbert space formulation of quantum mechanics, the
wave functions of Schrödinger’s wave mechanics are square-integrable functions
belonging to a Hilbert space that is isomorphic
1
, thus physically equivalent, to the
state space, and it is known as position or momentum space, depending on whether
the wave functions are expressed in terms of position or momentum, respectively. The
position and momentum spaces are also referred to as position and momentum
representations, respectively.
Apart from their significance in the historical development of quantum
mechanics, the position and momentum spaces play important role in the process of
teaching – thus of learning too – and also of applying the quantum theory to physical
systems, as it can be seen by referring to standard quantum mechanics textbooks
(Griffiths, 2005; Merzbacher, 1998; Sakurai & Napolitano, 2011).
However, as demonstrated in (Marshman & Singh, 2013 & 2015), students
face difficulties when practicing quantum mechanics in different spaces. Therefore, a
need exists for an alternative presentation of the quantum mechanical position and
momentum spaces. To this end, we reformulate the physical equivalence of the state
space to the position and momentum spaces in terms of a simple relation and provide
an intuitively geometric presentation of the latter spaces starting from the former
space. For simplicity and clarity, we examine a one-particle system, and the emphasis
is given on the one dimensional case
2
, i.e. the case where the particle moves on the
real line, as the generalization to three dimensions is straightforward.
From the State Space to the Position Space through a Simple Relation
We consider a particle moving, under the influence of a potential, on the real
line. The state space of our system is an abstract Hilbert space of Dirac kets. The
position space is then realized by taking projections of ket states on the directions – on
the axes – defined by the position eigenstates of the particle. Each position eigenstate
x
,
x
, defines a direction – an axis – by means of the projection operator
xx
.
We call this axis the
x
-axis. We note that the
x
-axis does not belong either to the
state space or to the position space. It belongs to a hyperspace, i.e. a bigger space, of
the state space (see below). In physical space, the
x
-axis corresponds to a point
x
on
the real line, along which the particle moves. Another position eigenstate, say
x
,
xx
, defines, by means of the projection operator
xx
, another axis, the
x
-axis,
1
Two Hilbert spaces
12
,HH
are isomorphic if there exists an isomorphism relating them, i.e. a linear mapping
U
from
1
H
to
2
H
that it is everywhere defined on
1
H
, it is onto on
2
H
, and it preserves the scalar product, i.e.
U
is a unitary operator
from
1
H
to
2
H
. Two isomorphic Hilbert spaces are considered as physically equivalent.
2
We note that the position and momentum spaces of a particle moving in one dimension are often referred to as one-dimensional
position and momentum spaces, respectively, and similarly for a particle moving in three dimensions, but this does not mean that
the two spaces are one dimensional (or three dimensional, respectively). They are infinite-dimensional, as is the state space.
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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which, in physical space, corresponds to the point
xx
on the real line, along which
the particle moves. The position of the particle is an observable quantity, thus the
position operator for the particle is Hermitian, and then its eigenstates belonging to
different eigenvalues are orthogonal (Dirac, 1947; Griffiths, 2005; Merzbacher, 1998).
As a result, the
x
-axis is orthogonal to the
x
-axis.
We consider an element of the state space, say the state
, and the state
ˆ
A
, where
ˆ
A
is an operator acting on the state space and
belongs to the
domain of
ˆ
A
. The states
and
ˆ
A
are respectively projected on the
x
-axis as
xx
and
ˆ
x A x
. The signed magnitudes of the two projections are,
respectively,
x
and
ˆ
xA
. We identify the element
x
as the value of the
position-space wave function at the point
x
, i.e.
( )
def.
xx
=
(1)
The element
ˆ
xA
is the projection of the state
ˆ
A
on the
x
-axis. The
state space is isomorphic to the position space, thus the description of the particle in
state space is equivalent to its description in position space.
We express the equivalence of the state space to the position space by the
following relation
( )
ˆˆ
x A A x x
=
(2)
where the operator
( )
ˆ
Ax
is the expression of the operator
ˆ
A
in position
space. Since the element
x
is the position-space wave function (1), (2) is also
written as
( ) ( )
ˆˆ
x A A x x
=
(3)
Thus, if the state-space operator
ˆ
A
is expressed, in position space, by the
operator
( )
ˆ
Ax
, then the state
ˆ
A
is described, in position space, by the wave
function
( ) ( )
ˆ
A x x
, where
( )
x
is the wave function describing the state
.
Position and Momentum in Position Space
The expression
( )
ˆ
xx
of the position operator
ˆ
x
in position space can be
derived by considering the element
ˆ
xx
, which, by means of (3), is written as
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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( ) ( )
ˆˆ
x x x x x
=
(4)
Since the position is an observable quantity, the position operator has a
complete set of eigenstates (Dirac, 1947; Griffiths, 2005; Merzbacher, 1998). Thus,
the set
x
x
spans the state space, and also consists of orthogonal eigenstates,
because the position operator is Hermitian. As a result, it holds the closure relation
ˆ
1dx x x
−
=
,
i.e. the sum of all projection operators
xx
,
x
, is equal to the identity
operator. Besides, the orthogonality of two arbitrary position eigenstates
x
and
x
is expressed by the relation
3
( )
x x x x
=−
,
where
( )
xx
−
is the delta function with support at
xx
=
.
The arbitrary position eigenstate
x
corresponds, in position space, to the
wave function
xx
, which is the delta function
( )
xx
−
. Since the state and
position spaces are isomorphic, the norms of
x
and
( )
xx
−
are equal. The norm
of
( )
xx
−
is infinite, since
( ) ( ) ( ) ( )
2
dx x x dx x x x x x x
− −
− = − − = − =
Then, the position eigenstates have infinite norm and thus they do not belong
to the state space, because, as we have mentioned, the state space contains states of
finite norm, which correspond to square-integrable position-space wave functions
describing physical states. This means that the position eigenstates are not physical
states. They are meant as generalized kets that span the state space as elements of a
hyperbasis, i.e. a basis belonging to a bigger space that contains the state space and is
accommodated in a construction called rigged Hilbert space (de la Madrid, 2005).
By means of the closure relation of the position eigenstates, the element
ˆ
xx
is written as
ˆ ˆ ˆ
x x x x dx x x dx x x x x
− −
==
,
3 The spectrum of the position operator is continuous, thus the place of the Kronecker delta, which is
used to express the orthogonality of discrete-basis states, has been taken by the delta function.
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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i.e.
ˆˆ
x x dx x x x x
−
=
(5)
Since
x
is a position eigenstate with eigenvalue
x
,
ˆ
x x x x
=
, and thus
( )
ˆ
x x x x x x x x x x x x
= = = −
Also, the element
x
is the wave function
( )
x
at
x
. Thus, (5) reads
( ) ( ) ( )
ˆ
x x dx x x x x x x
−
= − =
,
i.e.
( )
ˆ
x x x x
=
Comparing the last equation with (4) yields
( ) ( ) ( )
ˆ
x x x x x
=
,
and since the wave function is arbitrarily chosen,
( )
ˆ
x x x=
Thus, in position space, the position operator is the position coordinate. The
expression
( )
ˆ
px
of the momentum operator
ˆ
p
in position space can then be derived
by the canonical commutation relation for position and momentum
ˆˆ
,x p i=
(6)
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which, in position space, reads
( ) ( )
ˆˆ
,x x p x i=
(7)
The momentum operator
( )
ˆ
px
is linear and Hermitian. We observe that a
“solution” for
( )
ˆ
px
to (7) is
( )
ˆd
p x i dx
=−
(8)
If there exists another solution to (7), say
( )
ˆ
Px
, then
( ) ( ) ( ) ( )
ˆ
ˆ ˆ ˆ
,,x x p x x x P x
=
or
( ) ( ) ( )
ˆ
ˆˆ
,0x x p x P x
−=
,
i.e. the operator
( ) ( )
ˆ
ˆ
p x P x−
commutes with the position operator. Then,
since
( )
ˆ
x x x=
, the operator
( ) ( )
ˆ
ˆ
p x P x−
must be a function of
x
, i.e.
( ) ( ) ( )
ˆ
ˆ
p x P x f x−=
Besides, if
( )
px
is a momentum eigenfunction in position space, with
momentum
p
, then
( ) ( ) ( )
ˆ
p x p x pp x=
and
( ) ( ) ( )
ˆ
P x p x pp x=
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and thus, subtracting the two previous equations,
( ) ( )
0f x p x =
As an eigenfunction,
( )
px
cannot be the zero function, and then from the last
equation we derive that
( )
fx
is the zero function, and thus
( ) ( )
ˆˆ
P x p x=
, which
means that the solution (8) is unique.
To summarize, in position space, the position operator is the position
coordinate, while the momentum operator is the differential operator (8).
The momentum eigenfunctions
( )
px
in position space are then derived by
solving the momentum eigenvalue equation in position space, which, by means of (8),
reads
( ) ( )
i p x pp x
−=
(9)
and it is easily solved by separation of variables and integration, to give
( )
exp ipx
p x A
=
(10)
where
A
is a complex constant that does not depend either on
x
or on
p
4
.
Since
( )
px
is not square-integrable, as
( )
px
is constant, the constant
A
cannot be
calculated by a normalization condition.
To calculate the constant
A
, we proceed as follows: similarly to the wave
function (1), we write the momentum eigenfunction
( )
px
in position space as
( )
def.
p x x p=
(11)
where
p
is a momentum eigenstate. Since the wave function
( )
px
is not
square integrable, the ket
p
has infinite norm. As it happens with the position
eigenstates, the momentum eigenstates are also meant as generalized kets not
belonging to the state space, but since the momentum operator represents an
observable quantity, i.e. the momentum, the momentum eigenstates span the state
space and also, they are orthogonal, which means that
( )
p p p p
=−
.
4
We note that, in position space, the position
x
is a variable, while the momentum
p
is a parameter.
In momentum space, the opposite holds.
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Using the closure relation of the position eigenstates, the element
pp
is
written as
*
p p p dx x x p dx p x x p dx x p x p
− − −
= = =
,
where, in the last equality, we used the conjugate symmetry of the scalar
product (the asterisk denotes complex conjugation). Thus,
*
p p dx x p x p
−
=
,
By means of (11) and then (10), the last equality reads
( )
2exp i p p x
p p A dx
−
−
=
,
and then by the orthogonality of the eigenstates
p
and
p
, we obtain
( ) ( )
2exp i p p x
A dx p p
−
−
=−
(12)
To proceed, we’ll use the following integral representation of the delta
function
( ) ( )
1exp
2du ivu v
−
=
,
where
is a real parameter. Setting
pp
=−
, the previous equation reads
( )
( )
( )
1exp
2du i p p u p p
−
− = −
(13)
The delta function is even, i.e.
( ) ( )
p p p p
− = −
, thus comparing (12)
and (13) yields
( ) ( )
( )
21
exp exp
2
i p p x
A dx du i p p u
− −
−
=−
(14)
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Finally, changing the variable
x
to
yx=
, the integral on the left-hand side
reads
( )
( )
expdy i p p y
−
−
,
which is equal to the integral on the right-hand side of (14) times the reduced
Planck constant, as the integration variable is dummy and can be renamed to
u
. Thus,
(14) finally gives
1
2
A
=
,
and up to a constant phase, we end up to
1
2
A
=
The momentum eigenfunction (10) then reads
( )
1exp
2
ipx
px
=
(15)
or, using the definition (11),
1exp
2
ipx
xp
=
(16)
Besides, from the conjugate symmetry of the scalar product, we have
*
p x x p=
, and then, by means of (16), we obtain
1exp
2
ipx
px
=−
(17)
The element
px
is the projection of the arbitrary position eigenstate
x
on
the arbitrary momentum eigenstate
p
. Then, similarly to (11), we identify the
element
px
as the position eigenfunction
( )
xp
in momentum space, i.e.
( )
1exp
2
ipx
xp
=−
(18)
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The Momentum Space
Similarly to the position space, the momentum space of our particle is realized
by taking projections of kets on the directions – on the axes – defined by the
momentum eigenstates
p
,
p
, by means of the projection operators
pp
.
Then, similarly to (1), the momentum-space wave function
( )
p
describing a state
in momentum space is defined as
( )
def.
pp
=
(19)
As the position space, the momentum space is also physically equivalent to the
state space. Thus, similarly to (3), if
ˆ
A
is an operator acting on state space, then the
physical equivalence of the state space to the momentum space implies that
( ) ( )
ˆˆ
p A A p p
=
(20)
where the operator
( )
ˆ
Ap
is the expression of the operator
ˆ
A
in momentum
space.
We can now express the momentum-space wave function in terms of the
position-space wave function. Indeed, inserting the closure relation of the position
eigenstates on the right-hand side of (19), between the bra
p
and the ket
, we
obtain
( ) ( )
1exp
2
ipx
p p dx x x dx p x x dx x
− − −
= = = −
,
where, in the last equality, we used (1) and (17). Thus, the momentum-space
wave function is the Fourier transform of the position-space wave function, i.e.
( ) ( )
1exp
2
ipx
p dx x
−
=−
In the same way, inserting the closure relation of the momentum eigenstates
5
on the right-hand side of (1), and using (16) and (19), we write the position-space
wave function as the inverse Fourier transform of the momentum-space wave
function, i.e.
5
As the position eigenstates, the momentum eigenstates also satisfy a closure relation.
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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( ) ( )
1exp
2
ipx
x dp p
−
=
As we did to find the expression of the position operator in position space, the
expression
( )
ˆ
pp
of the momentum operator in momentum space can be derived by
considering the element
ˆ
pp
, where
is an arbitrary state of our particle.
Using (20), the previous element is written as
( ) ( )
ˆˆ
p p p p p
=
(21)
Also, using the closure relation of the momentum eigenstates, the element
ˆ
pp
is written as
ˆ ˆ ˆ
p p p p dp p p dp p p p p
− −
==
The ket
p
represents a momentum eigenstate, with momentum
p
, thus
ˆ
p p p p
=
, and the element
ˆ
ppp
reads
p p p
, and since the momentum
eigenstates are orthogonal, the last expression reads
( )
p p p
−
. Also, from (19), the
element
p
is the momentum-space wave function
( )
p
. Then, performing the
delta-function integration, we end up to
( )
ˆ
p p p p
=
Comparing the last equation with (21) and taking into account that the wave
function
( )
p
is arbitrary, we obtain
( )
ˆ
p p p=
,
i.e. in momentum space, the momentum operator is the momentum coordinate.
To derive the expression
( )
ˆ
xp
of the position operator in momentum space,
we can use the reasoning we employed to derive the momentum operator in position
space. Alternatively, we can use the position eigenfunctions we have already
calculated in (18). Thus, the position eigenvalue equation in momentum space reads
( )
ˆexp exp
ipx ipx
x p x
− = −
,
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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with a solution for
( )
ˆ
xp
( )
ˆd
x p i dp
=
It can be easily seen that the operators
i d dp
and
p
satisfy the canonical
commutation relation (6) in momentum space. Therefore, in momentum space, the
position operator is the differential operator
i d dp
, while the momentum operator is
the momentum coordinate.
Example
As an example, we’ll show that the stationary Schrödinger equation is the
energy eigenvalue equation in position space.
Considering a particle of mass
m
in a one-dimensional potential, its
Hamiltonian in state space reads
( )
2
ˆ
ˆˆ
2
p
H V x
m
=+
,
where
ˆˆ
,xp
are, respectively, the position and momentum operators. Then, the
energy eigenvalue equation for the particle reads
ˆEE
HE
=
,
where
E
is an eigenstate of energy
E
. Projecting both sides of the previous
equation on the axis defined by the position eigenstate
x
yields
( )
ˆE E E E
x H x E E x E x
= = =
,
where, in the last equality, we used the definition of the position-space wave
function (see eq. (1)). Thus
( )
ˆEE
x H E x
=
By means of (2), the element
ˆE
xH
reads
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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( ) ( ) ( )
ˆˆ
EE
H x x H x x
=
,
where
( )
ˆ
Hx
is the expression of the previous Hamiltonian in position space.
Thus
( ) ( ) ( )
ˆEE
H x x E x
=
(22)
In position space, the position operator is the position coordinate, while the
momentum operator is the differential operator
i d dx−
. Thus, the Hamiltonian of
the particle in position space is
( ) ( )
22
2
ˆ2d
H x V x
m dx
= − +
,
and (22) is then written as
( ) ( ) ( ) ( )
2
2E E E
x V x x E x
m
− + =
or
( ) ( )
( )
( )
2
20
EE
m
x E V x x
+ − =
,
which is the stationary Schrödinger equation (also known as the time-
independent Schrödinger equation).
The Three-Dimensional Case
For a particle moving in three dimensions, using the same reasoning as in the
one-dimensional case, the position space is realized by projecting ket states
on
the particle’s position eigenstates
r
, where
r
is a vector in
3
. The position-space
wave function is then defined as
( )
def.
rr
=
Similarly to (3), if
ˆ
A
is a state-space operator and the state
belongs to its
domain, then
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
48
( ) ( )
ˆˆ
r A A r r
=
,
where
( )
ˆ
Ar
is the expression of the operator
ˆ
A
in position space.
The orthogonality of the position eigenstates
r
and
r
is expressed by the
relation
( )
r r r r
=−
,
where, in Cartesian coordinates, the three-dimensional delta function is
( ) ( ) ( ) ( )
r r x x y y z z
− = − − −
The closure relation for the position eigenstates now reads
3ˆ
1d r r r
−
=
In three dimensions, the canonical commutation relations for position and
momentum read, in state space,
ˆˆ
[ , ]
i j ij
r p i
=
,
where the indices
,ij
take the values 1,2,3, with 1 standing for the coordinate
x
, 2 for
y
, and 3 for
z
.
Using the same reasoning as in the one-dimensional case, we find that the
expressions of the position and momentum operators in position space are,
respectively,
( ) ( )
ˆˆ
and r r r p r i= = −
,
where
is the del operator. That is, the position operator is the position
vector, while the momentum operator is the del operator times
i−
, which is an
obvious generalization from the one-dimensional case.
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49
The momentum eigenfunctions in position space are then derived by solving
the momentum eigenvalue equation
( ) ( )
i p r pp r− =
,
which, in Cartesian coordinates, reads
( ) ( ) ( )
( )
( )
ˆ ˆ ˆ ˆ ˆ ˆ
x y z x x y y z z
p r p r p r
i e e e p e p e p e p r
x y z
− + + = + +
(23)
where
ˆ ˆ ˆ
,,
x y z
e e e
are the unit vectors on the axes
,,x y z
, respectively. We note
that the momentum eigenfunctions are, actually, wave functions, thus they are scalar
functions. We can easily solve (23) by separating the variables, i.e. by writing the
momentum eigenfunction as
( ) ( ) ( ) ( )
p r X x Y y Z z=
Then, substituting into (23), we obtain three differential equations with the
same form as (9), thus the momentum eigenfunction
( )
pr
is the product of three
eigenfunctions with the form (15), i.e.
( )
1 1 1
exp exp exp
2 2 2
y
xz
ip y
ip x ip z
pr
=
or
( ) ( )
32
1exp
2
ip r
pr
=
(24)
We note that with the same reasoning, in
n
spatial dimensions, the momentum
eigenfunctions are, in position space,
( ) ( )
2
1exp
2nip r
pr
=
,
where
( )
1,..., n
r r r=
and
( )
1,..., n
p p p=
.
Since
( )
p r r p=
, (24) is also written as
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
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( )
32
1exp
2
ip r
rp
=
The description in momentum space is completely analogous. The momentum-
space wave function is defined as
( )
def.
pp
=
Similarly to (3), if
ˆ
A
is a state-space operator and the state
belongs to its
domain, then
( ) ( )
ˆˆ
p A A p p
=
,
where
( )
ˆ
Ap
is the expression of the operator
ˆ
A
in momentum space.
The orthogonality relation of the momentum eigenstates reads
( )
p p p p
=−
,
where, in Cartesian momentum coordinates,
( ) ( )
( )
( )
x x y y z z
p p p p p p p p
− = − − −
The momentum eigenstates satisfy the closure relation
3ˆ
1d p p p
−
=
The expressions of the position and momentum operators in momentum space
are, respectively,
( ) ( )
ˆˆ
and
p
r p i p p p= =
,
where
p
is the del operator in momentum, i.e. in Cartesian momentum
coordinates,
European J of Physics Education Volume 11 Issue 2 1309-7202 Konstantogiannis
51
ˆ ˆ ˆ
x y z
p p p p
x y z
e e e
p p p
= + +
Since
( ) ( )
**
r p p r r p p r= = =
,
the position eigenfunctions in momentum space are the complex conjugates of
the momentum eigenfunctions in position space, as it happens in one-dimension too.
Then, by means of (24),
( ) ( )
32
1exp
2
ip r
rp
=−
The relation between the position and momentum space wave functions
( )
r
and
( )
p
is derived in the same way as in the one-dimensional case, and we obtain
( ) ( ) ( )
3
32
1exp
2
ip r
p d r r
−
=−
,
i.e. the momentum-space wave function is the three-dimensional Fourier
transform of the position-space wave function. The three-dimensional inverse Fourier
transform then relates the position-space wave function to the momentum-space wave
function, i.e.
( ) ( ) ( )
3
32
1exp
2
ip r
r d p p
−
=
CONCLUSIONS
We have reformulated the physical equivalence of the quantum mechanical state
space to the position and momentum spaces in terms of a simple relation and we have
provided a presentation of the position and momentum spaces that is intuitively
geometric in that it is more constructive and less calculative than the presentations
found in standard quantum mechanics textbooks, and thus it is deprived of such
mathematical subtleties as the presence of the delta function derivative, which,
although well defined, is not well understood by physics students.
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By the present work, we hope to offer a perspective complementary to those
given in literature, which will help students to acquire functional knowledge of the
quantum mechanical position and momentum spaces.
REFERENCES
Aspect, A. & Villain, J. (2017). The birth of wave mechanics. Comptes Rendus
Physique, 18, 583-585.
de la Madrid, R. (2005). The role of the rigged Hilbert space in quantum mechanics.
European Journal of Physics, 26 (2), 287-312.
Dirac, P. A. M. (1947). The Principles of Quantum Mechanics, third edition.
Clarendon Press.
Gieres, F. (2000). Mathematical surprises and Dirac's formalism in quantum
mechanics. Reports on Progress in Physics, 63 (12), 1893.
Griffiths, D. J. (2005). Introduction to Quantum Mechanics, second edition. Pearson
Prentice Hall.
Hong, D., Wang, J., & Gardner, R. (2005). Real Analysis with an Introduction to
Wavelets and Applications, first edition. Academic Press.
Marshman, E. & Singh, C. (2013). Investigating Student Difficulties with Dirac
Notation. Paper presented at the 2013 Physics Education Research Conference.
Retrieved from arXiv:1510.01296.
Marshman, E. & Singh, C. (2015). Student difficulties with quantum states while
translating state vectors in Dirac notation to wave functions in position and
momentum representations. Paper presented at the 2015 Physics Education
Research Conference. Retrieved from arXiv:1509.04084.
Merzbacher, E. (1998). Quantum Mechanics, third edition. John Wiley & Sons, Inc.
Sakurai, J. J. & Napolitano, J. J. (2011). Modern Quantum Mechanics, second edition.
Pearson Education, Inc.
Van Hove, L. (1958). Von Neumann's contributions to quantum theory. Bulletin of the
American Mathematical Society, 64 (3), 95-99.
