Now returning to the discussion of the general Eq. (235), and comparing its last form with that of Eq. (236), we see that for a local operator in the coordinate representation, the eigenproblem (219) takes the form
ˆ A ( x) A ( x),
(4.248) Eigenproblem
A
A
in x-
representation
even if the operator A ˆ does not commute with the operator x ˆ . The most important case of this coordinate-representation form of the eigenproblem (68) is the familiar Eq. (1.60) for the eigenvalues En of the energy of a system with a time-independent Hamiltonian.
The operator locality also simplifies the expression for its expectation value. Indeed, plugging the closure relation in the form (231) into the general Eq. (125) twice (written in the first case for x and in the second case for x’), we get
ˆ
*
ˆ
A dx dx' ( t) x x A x' x' ( t) dx dx' ( x, t) x A x' ( x' , t)
.
(4.249)
Now, Eq. (236) reduces this result to just
A dx dx' *( x, t A ˆ
) ( x, t)
x x'
*( x, t ˆ
)
A
( x, t) dx
.
(4.250)
i.e. to Eq. (1.23), which had to be postulated in Chapter 1.
Finally, let us discuss the time evolution of the wavefunction, in the Schrödinger picture. For that, we may use Eq. (233) to calculate the (partial) time derivative of the wavefunction of some state :
i
i
x ( t) .
(4.251)
t
t
Since the coordinate operator x ˆ does not depend on time explicitly, its eigenstates x are stationary, and we can swap the time derivative and the time-independent bra-vector x. Now using the Schrödinger-picture equation (158), and then inserting the identity operator in the continuous form (220) of the closure relation, written for the coordinate eigenstates,
dx' x' x'
I ˆ
,
(4.252)
we may continue to develop the right-hand side of Eq. (251) as
x i
( t
ˆ
)
x H ( t
ˆ
)
dx' x H x' x' ( t
ˆ
)
dx' x H x' Ψ ( x' ) ,
(4.253)
t
Chapter 4
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If the Hamiltonian operator is local, we may apply Eq. (236) to the last expression, to get the familiar form (1.28) of the Schrödinger equation:
i
H ˆ
,
(4.254)
t
in which the coordinate representation of the operator H îs implied.
So, for the local operators that obey Eq. (236), we have been able to derive all the basic notions and postulates of the wave mechanics from the bra-ket formalism. Moreover, the formalism has allowed us to get the very useful equation (248) for an arbitrary local operator, which will be repeatedly used below. (In the first three chapters of this course, we have only used its particular case (1.60) for the Hamiltonian operator.)
Now let me deliver on my promise to develop a more balanced view at the monochromatic de Broglie waves (1), which would be more respectful to the evident r p symmetry of the coordinate and momentum. Let us do this for the 1D case when the wave may be represented as
px
( x) a exp
,
all
for
.
(4.255)
p
p
i
x
(For the sake of brevity, from this point to the end of the section, I am dropping the index x in the notation of the momentum – just as it was done in Chapter 2.) Let us have a good look at this function.
Since it satisfies Eq. (248) for the 1D momentum operator (245),
ˆ p p ,
(4.256)
p
p
p is an eigenfunction of that operator. But this means that we can also write Eq. (219) for the corresponding ket-vector:
p ˆ p p p ,
(4.257)
and according to Eq. (233), the wavefunction (255) may be represented as
( x) x p .
(4.258)
p
This expression is quite remarkable in its x p symmetry – which may be pursued further on.
Before doing that, however, we have to discuss the normalization of such wavefunctions. Indeed, in this case, the probability density w( x) (18) is constant, so that its integral
w( x) dx ( x) *
( x) dx
(4.259)
p
p
diverges if ap 0. Earlier in the course, we discussed two ways to avoid this divergence. One is to use a very large but finite integration volume – see Eq. (1.31). Another way is to work with wave packets of the type (2.20), possibly of a very large length and hence a very narrow spread of the momentum values.
Then the integral (259) may be required to equal 1 without any conceptual problem.
However, both these methods, convenient for the solution of many particular problems, violate the x p symmetry and hence are inconvenient for our current conceptual discussion. Instead, let us continue to identify the eigenvectors p and p of the momentum with the bra- and ket-vectors aA and Chapter 4
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aA of the general theory described at the beginning of this section. Then the normalization condition (222) becomes
p p' ( p p' ).
(4.260)
Inserting the identity operator in the form (252), with the integration variable x’ replaced by x, into the left-hand side of this equation, and using Eq. (258), we can translate this normalization rule to the wavefunction language:
*
dx p x x p' dx ( x) ( x) ( p p' ).
(4.261)
p
p'
For the wavefunction (255), this requirement turns into the following condition:
*
( p p' ) x
2
a a
exp i
dx a 2
( p p' ) ( p p' ),
(4.262)
p
p'
p
so that, finally, ap = ei/(2)1/2, where is an arbitrary (real) phase, and Eq. (255) becomes52
1
px
( x) x p
exp
(4.263)
p
i
2 1/2
Now let us represent an arbitrary wavefunction ( x) as a wave packet of the type (2.20), based on the wavefunctions (263), taking = 0 for the notation brevity, because the phase may be incorporated into the (generally, complex) envelope function ( p):
1
px
x-
( x)
( )exp
.
(4.264) representation:
2
p
i
1/ 2
dp
wavefunctions
From the mathematical point of view, this is just a 1D Fourier spatial transform, and its reciprocal is 1
px
p-
( p)
( )exp
.
(4.265) representation:
2
x
1/ 2
i
dx
wavefunctions
These expressions are completely symmetric, and represent the same wave packet; this is why the functions ( x) and ( p) are frequently called the reciprocal representations of a quantum state of the particle: respectively, its coordinate ( x-) and momentum ( p-) representations. Using Eq. (258), and Eq.
(263) with = 0, they may be recast into simpler forms,
( x) ( p) x p dp,
( p)
( x) p x dx
,
(4.266)
in which the inner products satisfy the basic postulate (14) of the bra-ket formalism:
1
px
*
p x
.
(4.267)
exp i
x p
2
1/ 2
52 Repeating such calculation for each Cartesian component of a plane monochromatic wave of arbitrary dimensionality d, we get p = (2)– d/2exp{ i(pr/ + )}.
Chapter 4
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Next, we already know that in the x-representation, i.e. in the usual wave mechanics, the coordinate operator x îs reduced to the multiplication by x, and the momentum operator is proportional to the partial derivative over the coordinate:
x-
representation:
operators
ˆ x
x,
ˆ p
i
.
(4.268)
in x
in x
x
It is natural to guess that in the p-representation, the expressions for operators would be reciprocal: p-
representation:
ˆ x
i
,
ˆ p
p,
(4.269)
in p
in
operators
p
p
with the only difference of one sign, which is due to the opposite signs of the Fourier exponents in Eqs.
(264) and (265). The proof of Eqs. (269) is straightforward; for example, acting by the momentum operator on the arbitrary wavefunction (264), we get
1
px
1
px
ˆ p ( x) i
( x)
(4.270)
x
p
i
i
dp
p
p
i
dp
2
( )
exp
1/ 2
x
2
( ) exp
,
1/ 2
and similarly for the operator x âcting on the function ( p). Comparing the final form of Eq. (270) with the initial Eq. (264), we see that the action of the operators (268) on the wavefunction (i.e. the state’s x-representation) gives the same results as the action of the operators (269) on the function (i.e. its p-
representation).
It is illuminating to have one more, different look at this coordinate-momentum duality. For that, notice that according to Eqs. (82)-(84), we may consider the bracket x p as an element of the (infinite-size) matrix Uxp of the unitary transform from the x-basis to the p-basis. Let us use this fact to derive the general operator transform rule that would be a continuous version of Eq. (92). Say, we want to calculate the general matrix element of some operator, known in the x-representation, in the p-
representation:
p A ˆ p' .
(4.271)
Inserting two identity operators (252), written for x and x’, into this bracket, and then using Eq. (258) and its complex conjugate, and also Eq. (236) (again, valid only for space-local operators!), we get ˆ
ˆ
*
ˆ
p A p' dx dx' p x x A x' x' p' dx dx' ( x) x A x' ( x' )
p
p'
1
px
p'x'
1
(4.272)
ˆ
px ˆ
p'x
dx dx' exp
i
( x x' ) A exp i
dx exp
i
A exp i
.
2
2
As a sanity check, for the momentum operator itself, this relation yields:
1
px
p'x
p'
( p' p) x
p ˆ p p'
dx exp
i
i
exp i
exp i
dx p' ( p' p). (4.273)
2
x
2
Due to Eq. (257), this result is equivalent to the second of Eqs. (269).
From a thoughtful reader, I anticipate the following natural question: why is the momentum representation used much less frequently than the coordinate representation – i.e. wave mechanics? The answer is purely practical: besides the important special case of the 1D harmonic oscillator (to be Chapter 4
Page 47 of 52
QM: Quantum Mechanics
revisited in Sec. 5.4), in most systems the orbital-motion Hamiltonian (237) is not x p symmetric, with the potential energy U(r) typically being a more complex function than the kinetic energy p 2/2 m.
Because of that, it is easier to analyze such systems treating such potential energy operator just a c-
number multiplier, as it is in the coordinate representation – as it was done in Chapters 1-3.
The most significant exception from this practice is the motion in a periodic potential in presence of a coordinate-independent external force F( t). As was discussed in Secs. 2.7 and 3.4, in such periodic systems the eigenenergies En(q), playing the role of the effective kinetic energy of the particle, may be rather involved functions of its quasimomentum q, while its effective potential energy U ef = –F( t)r, due to the additional force F( t), is a very simple function of coordinates. This is why detailed analyses of the quantum effects briefly discussed in Sec. 2.8 (the Bloch oscillations, etc.) and also such statistical phenomena as drift, diffusion, etc.,53 in solid-state theory are typically based on the momentum (or rather quasimomentum) representation.
4.8. Exercise problems
4.1. Prove that if A ând B âre linear operators, and C is a c-number, then: (i) A ˆ† † A ˆ;
(ii) ˆ
CA†
* †
ˆ
C A ;
(iii) ˆ ˆ B
A †
† †
ˆ ˆ
B A ;
(iv) the operators
†
ˆ ˆ A
A
and A ˆ† A âre Hermitian.
4.2. Prove that for any linear operators ˆ ˆ ˆ
ˆ
,
A B, C
and
,
D,
ˆ B
A ˆ C ˆ
, D ˆ ˆ
A B ˆ C ˆ
, D ˆ ˆ
C
A ˆ B ˆ ˆ
,
D A ˆ C ˆ
, ˆ B
D ˆ C ˆ A ˆ D ˆ
, B ˆ .
4.3. Calculate all possible binary products j j’ (for j, j’ = x, y, z) of the Pauli matrices, defined by Eqs. (105), and their commutators and anticommutators (defined similarly to those of the corresponding operators). Summarize the results, using the Kronecker delta and Levi-Civita permutation symbols.54
4.4. Calculate the following expressions,
(i) (c) n, and then
(ii) ( b I + c) n,
for the scalar product c of the Pauli matrix vector n x x + n y y + n z z by an arbitrary c-number geometric vector c, where n 0 is an integer, and b is an arbitrary scalar c-number.
Hint: For task (ii), you may like to use the binomial theorem,55 and then transform the result in a way enabling you to use the same theorem backward.
53 In this series, a brief discussion of these effects may be found in SM Chapter 6.
54 See, e.g., MA Eqs. (13.1) and (13.2).
55 See, e.g. MA Eq. (2.9).
Chapter 4
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4.5. Use the solution of the previous problem to derive Eqs. (2.191) for the transparency T of a system of N similar, equidistant, delta-functional potential barriers.
4.6. Use the solution of Problem 4(i) to spell out the following matrix: exp{ i n}, where is the 3D vector (117) of the Pauli matrices, n is a c-number geometric vector of unit length, and is a c-
number scalar.
4.7. Use the solution of Problem 4(ii) to calculate exp{A}, where A is an arbitrary 22 matrix.
4.8. Express all elements of the matrix B exp{A} explicitly via those of the 22 matrix A.
Spell out your result for the following matrices:
a a
i i
A
,
A '
,
a a
i i
with real a and .
4.9. Prove that for arbitrary square matrices A and B,
Tr (
)
AB Tr (BA) .
Is each diagonal element ( AB) jj necessarily equal to ( BA) jj?
4.10. Calculate the trace of the following 22 matrix:
A a σb σc σ,
where is the Pauli matrix vector, while a, b, and c are arbitrary c-number vectors.
4.11. Prove that the matrix trace of an arbitrary operator does not change at its arbitrary unitary transformation.
4.12. Prove that for any two full and orthonormal bases { u} and { v} of the same Hilbert space, Tr u v
v u
j
j'
.
j'
j
4.13. Is the 1D scattering matrix S, defined by Eq. (2.124), unitary? What about the 1D transfer matrix T defined by Eq. (2.125)?
4.14. Calculate the trace of the following matrix:
exp ia
σ
exp ib
σ ,
where is the Pauli matrix vector, while a and b are c-number geometric vectors.
4.15. Prove the following vector-operator identity:
σ ˆrσ ˆp I ˆr ˆp iσ ˆr ˆp, where is the Pauli matrix vector, and I is the 22 identity matrix.
Chapter 4
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Hint: Take into account that the vector operators rând pâre defined in the orbital-motion Hilbert space, different from that of the Pauli vector-matrix , and hence commute with it – even though they do not commute with each other.
4.16. Let Aj be eigenvalues of some operator A ˆ . Express the following two sums,
,
,
1
A
A 2
j
2
j
j
j
via the matrix elements Ajj’ of this operator in an arbitrary basis.
4.17. Calculate of a spin–½ in the quantum state with the following ket-vector:
z
const ,
where (, ) and (, ) are the eigenstates of the Pauli matrices z and x, respectively.
Hint: Double-check whether your solution is general.
4.18. A spin-½ is fully polarized in the positive z-direction. Calculate the probabilities of the alternative outcomes of a perfect Stern-Gerlach experiment with the magnetic field oriented in an arbitrary different direction, performed on a particle in this spin state.
4.19. In a certain basis, the Hamiltonian of a two-level system is described by the matrix
E
0
1
H
,
with E E
,
1
2
0
E
2
while the operator of some observable A of this system, by the matrix
1
1
A
.
1
1
For the system’s state with the energy definitely equal to E 1, find the possible results of measurements of the observable A and the probabilities of the corresponding measurement outcomes.
4.20. Certain states u 1,2,3 form an orthonormal basis of a system with the following Hamiltonian ˆ
H u u u u u u
1
2
2
3
3
1
,
h.c.
where is a real constant, and h.c. means the Hermitian conjugate of the previous expression. Calculate its stationary states and energy levels. Can you relate this system to any other(s) discussed earlier in the course?
4.21. Guided by Eq. (2.203), and by the solutions of Problems 3.11 and 4.20, suggest a Hamiltonian describing particle’s dynamics in an infinite 1D chain of similar potential wells in the tight-binding approximation, in the bra-ket formalism. Verify that its eigenstates and eigenvalues correspond to those discussed in Sec. 2.7.
Chapter 4
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4.22. Calculate eigenvectors and eigenvalues of the following matrices:
0 0 0 1
0 1 0
0 0 1 0
A 1 0 1,
B
0 1 0 0
0 1 0
1 0 0 0
4.23. A certain state is an eigenstate of each of two operators, A ând B ˆ . What can be said about the corresponding eigenvalues a and b, if the operators anticommute?
4.24. Derive the differential equation for the time evolution of the expectation value of an observable, using both the Schrödinger picture and the Heisenberg picture of quantum dynamics.
4.25. At t = 0, a spin-½ whose interaction with an external field is described by the Hamiltonian H ˆ c ˆσ c σ ˆ c σ ˆ c σ ˆ
x
x
y
y
z
z
(where cx,y,z are real c-number constants, and ˆ
are the Pauli operators), was in the state , one of the
x, y, z
two eigenstates of ˆ . In the Schrödinger picture, calculate the time evolution of:
z
(i) the ket-vector of the spin (in any time-independent basis you like),
(ii) the probabilities to find the spin in the states and , and
(iii) the expectation values of all three Cartesian components ( ˆ S , etc.) of the spin vector.
x
Analyze and interpret the results for the particular case cy = cz = 0.
Hint: Think about the best basis to use for the solution.
4.26. For the same system as in the previous problem, use the Heisenberg picture to calculate the time evolution of:
(i) all three Cartesian components of the spin operator ˆS ( t), and
H
(ii) the expectation values of the spin components.
Compare the latter results with those of the previous problem.
4.27. For the same system as in the two last problems, calculate matrix elements of the operator
în the basis of the stationary states of the system.
z
4.28. In the Schrödinger picture of quantum dynamics, certain three operators satisfy the following commutation relation:
A ˆ B ˆ, C ˆ .
What is their relation in the Heisenberg picture, at a certain time instant t?
4.29. Prove the Bloch theorem given by either Eq. (3.107) or Eq. (3.108).
Chapter 4
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Hint: Consider the translation operator T ˆ defined by the following result of its action on an R
arbitrary function f(r):
ˆ
T f (r) f (r R) ,
R
for the case when R is an arbitrary vector of the Bravais lattice (3.106). In particular, analyze the commutation properties of this operator, and apply them to an eigenfunction (r) of the stationary Schrödinger equation for a particle moving in the 3D periodic potential described by Eq. (3.105).
4.30. A constant force F is applied to an (otherwise free) 1D particle of mass m. Calculate the stationary wavefunctions of the particle in:
(i) the coordinate representation, and
(ii) the momentum representation.
Discuss the relation between the results.
4.31. Use the momentum representation to re-solve the problem discussed at the beginning of Sec. 2.6, i.e. calculate the eigenenergy of a 1D particle of mass m, localized in a very short potential well of “weight” W.
4.32. The momentum representation of a certain operator of 1D orbital motion is p-1. Find its coordinate representation.
4.33.* For a particle moving in a 3D periodic potential, develop the bra-ket formalism for the q-
representation, in which a complex amplitude similar to aq in Eq. (2.234) (but generalized to 3D and all energy bands) plays the role of the wavefunction. In particular, calculate the operators r and v in this representation, and use the result to prove Eq. (2.237) for the 1D case in the low-field limit.
4.34. A uniform, time-independent magnetic field B is induced in one semi-
B
space, while the other semi-space is field-free, with a sharp, plane boundary between these two regions. A monochromatic beam of non-relativistic, electrically-neutral spin-½ particles with a gyromagnetic ratio 0,56 in a certain spin state and
with a kinetic energy E, is incident on this boundary, from the field-free side, under E
angle – see figure on the right. Calculate the coefficient of particle reflection from the boundary.
56 The fact that may be different from zero even for electrically-neutral particles, such as neutrons, is explained by the Standard Model of the elementary particles, in which a neutron “consists” (in a broad sense of the word) of three electrically-charged quarks with zero net charge.
Chapter 4
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Chapter 5. Some Exactly Solvable Problems
The objective of this chapter is to describe several relatively simple but very important applications of the bra-ket formalism, including a few core problems of wave mechanics we have already started to discuss in Chapters 2 and 3.
5.1. Two-level systems
The discussion of the bra-ket formalism in the previous chapter was peppered with numerous illustrations of its main concepts on the example of “spins-½” – systems with the smallest non-trivial (two-dimensional) Hilbert space, in which the bra- and ket-vectors of an arbitrary quantum state may be represented as a linear superposition of just two basis vectors, for example
,
(5.1)
where the states and were defined as the eigenstates of the Pauli matrix z – see Eq. (4.105). For the genuine spin-½ particles, such as electrons, placed in a z-oriented time-independent magnetic field, these states are the stationary “spin-up” and “spin-down” stationary states of the Pauli Hamiltonian (4.163), with the corresponding two energy levels (4.167). However, an approximate but reasonable quantum description of some other important systems may also be given in such Hilbert space.
For example, as was discussed in Sec. 2.6, two weakly coupled space-localized orbital states of a spin-free particle are sufficient for an approximate description of its quantum oscillations between two potential wells. A similar coupling of two traveling waves explains the energy band splitting in the weak-potential approximation of the band theory – Sec. 2.7. As will be shown in the next chapter, in systems with time-independent Hamiltonians, such situation almost unavoidably appears each time when two energy levels are much closer to each other than to other levels. Moreover, as will be shown in Sec. 6.5, a similar truncated description is adequate even in cases when two levels En and En’ of an unperturbed system are not close to each other, but the corresponding states become coupled by an applied ac field of a frequency very close to the difference ( En – En’ )/. Such two-level systems (alternatively called “spin-½-like” systems) are nowadays the focus of additional attention in the view of prospects of their use for quantum information processing and encryption.1 This is why let me spend a bit more time reviewing the main properties of an arbitrary two-level system.
First, the most general form of the Hamiltonian of a two-level system is represented, in an arbitrary basis, by a 22 matrix
H
H
H
11
12
.
(5.2)
H
H
21
22
According to the discussion in Secs. 4.3-4.5, since the Hamiltonian operator has to be Hermitian, the diagonal elements of the matrix H have to be real, and its off-diagonal elements be complex conjugates 1 In the last context, to be discussed in Sec. 8.5, the two-level systems are usually called qubits.
© K. Likharev
QM: Quantum Mechanics
of each other: H 21 = H 12*. As a result, we may not only represent H as a linear combination (4.106) of the identity matrix and the Pauli matrices but also reduce it to a more specific form:
b c
c ic b c
c
H b I
z
x
y
z
c σ
,
c c ic ,
(5.3)
x
y
c ic
b c
c
b c
x
y
z
z
where the scalar b and the Cartesian components of the vector c are real c-number coefficients: H H
H H
H H
H H
11
22
b
,
12
21
c
Re H ,
21
12
c
Im H ,
11
22
c
. (5.4)
2
x
2
21
y
2
21
i
z
2
If such Hamiltonian does not depend on time, the corresponding characteristic equation (4.103) for the system’s energy levels E,
b c E
c
z
0,
(5.5)
c
b c
E
z
is a simple quadratic equation, with the following solutions:
1/ 2
2
1/ 2
1/ 2
H H
H H
2
E b c b c c
c
b
c
c
c
H
. (5.6)
2
z
2
2
2
x
y
z
11
22
11
22
21
2
2
The parameter b ( H 11 + H 22)/2 evidently gives the average energy E(0) of the system, which does not contribute to the level splitting
1/
1/ 2
2
2
E E E 2 c 2
2 2 2
c c c
.
(5.7)
x
y
z
H H
4
H
11
22
2
21
So, the splitting is a hyperbolic function of the coefficient cz ( H 11 – H 22)/2. A plot of this function is the famous level-anticrossing diagram (Fig. 1), which has already been discussed in Sec. 2.7 in the particular context of the weak-potential limit of the 1D band theory.
(0)
E E
E
H
21
0
H H
11
22
c
z
H
2
21
E
Fig. 5.1. The level-anticrossing diagram
for an arbitrary two-level system.
The physics of the diagram becomes especially clear if the two states of the basis used to spell out the matrix (2), may be interpreted as the stationary states of two potentially independent subsystems, with the energies, respectively, H 11 and H 22. (For example, in the case of two weakly coupled potential wells discussed in Sec. 2.6, these are the ground-state energies of two distant wells.) Then the off-diagonal elements c
*
– H 12 and c+ H 21 = H 12 describe the subsystem coupling, and the level Chapter 5
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Essential Graduate Physics
QM: Quantum Mechanics
anticrossing diagram shows how do the eigenenergies of the coupled system depend (at fixed coupling) on the difference of the subsystem energies. As was already discussed in Sec. 2.7, the most striking feature of the diagram is that any non-zero coupling c
2
2
( cx + cy )1/2 changes the topology of the
eigenstate energies, creating a gap of the width E.
As it follows from our discussions of particular two-level systems in Secs. 2.6 and 4.6, their dynamics also has a general feature – the quantum oscillations. Namely, if we put any two-level system into any initial state different from one of its eigenstates , and then let it evolve on its own, the probability of its finding the system in any of the “partial” states exhibits oscillations with the frequency E
E E
2 c ,
(5.8)
lowest at the exact subsystem symmetry ( cz = 0, i.e. H 11 = H 22), when it is proportional to the coupling strength: min = 2 c/ 2 H 12/ = 2 H 21/.
In the case discussed in Sec. 2.6, these are the oscillations of a particle between the two coupled potential wells (or rather of the probabilities to find it in either well) – see, e.g., Eqs. (2.181). On the other hand, for a spin-½ particle in an external magnetic field, these oscillations take the form of spin precession in the plane normal to the field, with periodic oscillations of its Cartesian components (or rather their expectation values) – see, e.g., Eqs. (4.173)-(4.174). Some other examples of the quantum oscillations in two-level systems may be rather unexpected; for example, the ammonium molecule NH3
(Fig. 2) has two symmetric states that differ by the inversion of the nitrogen atom relative to the plane of the three hydrogen atoms, which are weakly coupled due to quantum-mechanical tunneling of the nitrogen atom through the plane of the hydrogen atoms.2 Since for this particular molecule, in the absence of external fields, the level splitting E corresponds to an experimentally convenient frequency
/2 24 GHz, it played an important historic role at the initial development of the atomic frequency standards and microwave quantum generators ( masers) in the early 1950s,3 which paved the way toward laser technology.
N
nm
0.102
107.8
H
H
H
Fig. 5.2. An ammonia molecule and its inversion.
Now let us now discuss a very convenient geometric representation of an arbitrary state of (any!) two-level system. As Eq. (1) shows, such state is completely described by two complex 2 Since the hydrogen atoms are much lighter, it would be fairer to speak about the tunneling of their triangle around the (nearly immobile) nitrogen atom.
3 In particular, these molecules were used in the demonstration of the first maser by C. Townes’ group in 1954.
Chapter 5
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coefficients ( c-numbers) – say, and . If the vectors of the basis states and are normalized, then these coefficients must obey the following restriction:
W
.
(5.9)
*
*
2
2
*
*
1
This requirement is automatically satisfied if we take the moduli of and equal to the sine and cosine of the same (real) angle. Thus we may write, for example,
i
i(
)
cos e , sin e
.
(5.10)
2
2
Moreover, according to the general Eq. (4.125), if we deal with just one system,4 the common phase factor exp{ i} drops out of the calculation of any expectation value, so that we may take = 0, and Eq.
(10) is reduced to
cos , sin
i
e
Bloch
.
(5.11)
2
2
sphere
representation
The reason why the argument of these sine and cosine functions is usually taken in the form /2, becomes clear from Fig. 3a: Eq. (11) conveniently maps each state of a two-level system on a certain representation point on a unit-radius Bloch sphere,5 with the polar angle and the azimuthal angle .
z
(a)
z
(b)
z
(c)
y
c
y
c
0
x
0
x
0
x
Fig. 5.3. The Bloch sphere: (a) the representation of an arbitrary state (solid red point) and the eigenstates of the Pauli matrices (dotted points), and (b, c) the two-level system’s evolution: (b) in a constant “field” c directed along the z-axis, and (c) in a field of arbitrary orientation.
In particular, the basis state , described by Eq. (1) with = 1 and = 0, corresponds to the North Pole of the sphere ( = 0), while the opposite state , with = 0 and = 1, to its South Pole (
= ). Similarly, the eigenstates and of the matrix x, described by Eqs. (4.122), i.e. having =
4 If you need a reminder of why this condition is crucial, please revisit the discussion at the end of Sec. 1.6. Note also that the mutual phase shifts between different qubits are important, in particular, for quantum information processing (see Sec. 8.5 below), so that most discussions of these applications have to start from Eq. (10) rather than Eq. (11).
5 This representation was suggested in 1946 by the same Felix Bloch who has pioneered the energy band theory discussed in Chapters 2-3.
Chapter 5
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1/2 and = 1/2, correspond to the equator ( = /2) points with, respectively, = 0 and = .
Two more special points (denoted in Fig. 3a as ⊙ and ) are also located on the sphere’s equator, at =
/2 and = /2; it is easy to check that they correspond to the eigenstates of the matrix y (in the same z-basis).
To understand why such mutually perpendicular location of these three special point pairs on the Bloch sphere is not occasional, let us plug Eqs. (11) into Eqs. (4.131)-(4.133) for the expectation values of the spin-½ components. In terms of the Pauli vector operator (4.117), S/(/2), the result is
sin cos,
sin sin,
cos ,
(5.12)
x
y
z
showing that the radius vector of any representation point is just the expectation value of .
Now let us use Eq. (3) to see how does the representation point moves in various cases, ignoring the term b I – which, again, describes the offset of the total energy of the system relative to some reference level, and does not affect its dynamics. First of all, according to Eq. (4.158), in the case c = 0
(when the Hamiltonian operator turns to zero, and hence the state vectors do not depend on time) the point does not move at all, and its position is determined by initial conditions, i.e. by the system’s preparation. If c 0, we may re-use some results of Sec. 4.6, obtained for the Pauli Hamiltonian (4.163a), which coincides with Eq. (3) if6
c
B .
(5.13)
2
In particular, if the field B, and hence the vector c, is directed along the z-axis and is time-independent, Eqs. (4.170) and (4.173)-(4.174) show that the representation point on the Bloch sphere rotates within a plane normal to this axis (see Fig. 3b) with the angular velocity
d
2 c
z
B
.
(5.14)
z
dt
Almost evidently, since the selection of the coordinate axes is arbitrary, this picture should remain valid for any orientation of the vector c, with the representation point rotating, on the Bloch sphere, around it direction, with the angular speed = 2 c/ – see Fig. 3c. This fact may be proved using any picture of the quantum dynamics, discussed in Sec. 4.6. Actually, the reader may already have done that by solving Problems 4.25 and 4.26, just to see that even for the particular, simple initial state of the system (), the final results for the Cartesian components of the vector are somewhat bulky.
However, this description may be readily simplified, even for arbitrary time dependence of the “field”
vector c( t) in Eq. (3), using the (geometric) vector language.
Indeed, let us rewrite Eq. (3) (again, with b = 0) in the operator form,
ˆ
H c t σˆ ,
(5.15)
valid in an arbitrary basis. According to Eq. (4.199), the corresponding Heisenberg equation of motion for the j th Cartesian components of the vector-operator σˆ (which does not depend on time explicitly, so that ˆ
/ t 0 ) is
6 This correspondence justifies using the use of term “field” for the vector c.
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3
3
i ˆ
H
c t σ
c t
c t .
(5.16)
j
ˆ ˆ,
j
ˆ , j ˆ ˆ , j ˆ
ˆ , ˆ
j'
j' j' j
j'
j'1
j'1
Now using the commutation relations (4.155), which remain valid in any basis and in any picture of time evolution,7 we get
3
i ˆ
2 i
c t
,
(5.17)
j
ˆ
j'
j"
jj'j"
j'1
where j” is the index, or the same set {1, 2, 3}, complementary to j and j’ ( j” j, j’), and jj’j” is the Levi-Civita symbol.8 But it is straightforward to verify that the usual vector product of two 3D vectors may be represented in a similar Cartesian-component form:
n
n
n
1
2
3
3
a b a
a
a
a b
,
(5.18)
j
1
2
3
j' j" jj'j"
j'1
b
b
b
1
2
3 j
As a result, Eq. (17) may be rewritten in a vector form – or rather several equivalent forms: 2
i ˆ
2 i
(5.19)
j
c t ˆσ ,
i.e. iˆσ 2 ic
j
t ˆσ,
or ˆσ c t ˆσ,
or ˆσ Ω t ˆσ,
where the vector is defined as
Ω t c
2 t
(5.20)
– an evident generalization of Eq. (14).9 As we have seen in Sec. 4.6, any linear relation between two Heisenberg operators is also valid for the expectation values of the corresponding observables, so that the last form of Eq. (19) yields:
σ Ω t σ .
(5.21)
But this is the well-known kinematic formula10 for the rotation of a constant-length classical 3D
vector around the instantaneous direction of the vector ( t), with the instantaneous angular velocity
( t). So, the time evolution of the representation point on the Bloch sphere is quite simple, especially in the case of a time-independent c, and hence – see Fig. 3c.11 Note that it is sufficient to turn off the field to stop the precession instantly. (Since Eq. (21) is the first-order differential equation, the 7 Indeed, if some three operators in the Schrödinger picture are related as [ ˆ
ˆ
A , B ] = ˆ
C , then according to Eq.
S
S
S
(4.190), in the Heisenberg picture:
ˆ
ˆ
† ˆ
[ A , B ] [ ˆ u A ˆ u, ˆ† ˆ
† ˆ
u B ˆ u] ˆ u A ˆ u ˆ† ˆ
u B ˆ u ˆ† ˆ
† ˆ
† ˆ
u B ˆ u ˆ u A ˆ u ˆ
ˆ
† ˆ
ˆ
u [ A , B ] ˆ u ˆ u C ˆ u C .
H
H
H
H
H
H
H
H
S
S
S
H
8 See, e.g., MA Eq. (9.2). Note that in Eqs. (17)-(18) and similar expressions below, the condition j” j, j’ may be (and frequently is) replaced by the summation over not only j’, but also j” , in their right-hand sides.
9 It is also easy to verify that in the particular case = n z, Eqs. (19) are reduced, in the z-basis, to Eqs. (4.200) for the spin-½ vector matrix S = (/2).
10 See, e.g., CM Sec. 4.1, in particular Eq. (4.8).
11 The bulkiness of the solutions of Problems 4.25 and 4.26 (which were offered just as useful exercises in quantum dynamic formalisms) reflects the awkward expression of the resulting circular motion of the vector
(see Fig. 3c) via its Cartesian components.
Chapter 5
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representation point has no effective inertia.12) Hence, changing the direction and the magnitude of the effective external field, it is possible to drive the representation point of a two-level system from any initial position to any final position on the Bloch sphere, i.e. make the system take any of its possible quantum states.
In the particular case of a spin-½ in a magnetic field B( t), it is more customary to use Eqs. (13) and (20) to rewrite Eq. (21) as the following equation for the expectation value of the spin vector S =
(/2):
S S B t.
(5.22)
As we know from the discussion in Chapter 4, such a classical description of the spin’s evolution does not give a full picture of the quantum reality; in particular, it does not describe the possible large uncertainties of its components – see, e.g., Eqs. (4.135). The situation, however, is different for a collection of N >> 1 similar, non-interacting spins, initially prepared to be in the same state – for example by polarizing all spins with a strong external field B0, at relatively low temperatures T, with k B T << B0. (A practically important example of such a collection is a set of nuclear spins in macroscopic condensed-matter samples, where the spin interaction with each other and the environment is typically very small.) For such a collection, Eq. (22) is still valid, while the relative uncertainty of the resulting sample’s magnetization M = nm = nS (where n N/ V is the spin density) is proportional to 1/ N 1/2 << 1. Thus, the evolution of magnetization may be described, with good precision, by the essentially classical equation (valid for any spin, not necessarily spin-½):
M
M
B t.
(5.23)
This equation, or the equivalent set of three Bloch equations 13 for its Cartesian components, with the right-hand side augmented with small terms describing the effects of dephasing and relaxation (to be discussed in Chapter 7), is used, in particular, to describe the magnetic resonance, taking place when the frequency (4.164) of the spin’s precession in a strong dc magnetic field approaches the frequency of an additionally applied (and usually weak) ac field.14
5.2. The Ehrenfest theorem
In Sec. 4.7, we have derived all the basic relations of wave mechanics from the bra-ket formalism, which will also enable us to get some important additional results in that area. One of them is a pair of very interesting relations, together called the Ehrenfest theorem. To derive them, for the simplest case of 1D orbital motion, let us calculate the following commutator:
ˆ x, ˆ 2
p
(5.24)
x
ˆ x ˆ p ˆ p
ˆ p ˆ p ˆ.
x
x
x
x
x
Let us apply the commutation relation (4.238) in the following form:
ˆˆ p
x
ˆ p ˆ x i I,ˆ
(5.25)
x
x
12 This is also true for the classical angular momentum L at its torque-induced precession – see, e.g., CM Sec. 4.5.
13 They were introduced by F. Bloch in the same 1946 paper as the Bloch-sphere representation.
The quantum theory of this effect will be discussed in the next chapter.
Chapter 5
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to the first term of the right-hand side of Eq. (24) twice, with the goal to move the coordinate operator to the rightmost position:
ˆˆ p
x
ˆ p p x i I p p xp i p p p x i I i p p p x i p (5.26)
x
x
ˆ ˆ ˆ
x
ˆ ˆ ˆˆ
ˆ
ˆ
x
x
x
x
x ˆ
ˆ
ˆ
x
ˆ ˆ ˆ ˆ 2 ˆ .
x
x
x
x
The first term of this result cancels with the last term of Eq. (24), so that the commutator becomes quite simple:
ˆ x, ˆ2 p i p
(5.27)
x
2 ˆ .
x
Let us use this equality to calculate the Heisenberg-picture equation of motion of the operator x ˆ , by applying the general Heisenberg equation (4.199) to the 1D orbital motion described by the Hamiltonian (4.237), but possibly with a more general, time-dependent potential energy U: d ˆ x
1
1
ˆ 2
p
ˆ ˆ
x, H
ˆ
x, x U ( ˆ x, t) .
(5.28)
dt
i
i
2
m
The potential energy operator is a function of the coordinate operator and hence, as we know, commutes with it. Thus, the right-hand side of Eq. (28) is proportional to the commutator (27), and we get Heisenberg
d ˆ x
ˆ p
equation
x
.
(5.29) for
dt
m
coordinate
In this operator equality, we readily recognize the full analog of the classical relation between the particle’s momentum and is velocity.
Now let us see what a similar procedure gives for the momentum’s derivative:
d ˆ px
1
1
ˆ 2
p
ˆ ˆ
p , H
(5.30)
x
ˆ p , x U(ˆ x, t) .
x
dt
i
i
2
m
The kinetic energy operator commutes with the momentum operator and hence drops from the right-hand side of this equation. To calculate the remaining commutator of the momentum and potential energy, let us use the fact that any smooth (infinitely differentiable) function may be represented by its Taylor expansion:
k
1 U
k
U ( x ˆ, t)
x ˆ ,
(5.31)
k
0
! ˆ
k
k
x
where the derivatives of U may be understood as c-numbers (evaluated at x = 0, and the given time t), so that we may write
1 k
U
1
U
k
k
p ˆ , U ( x ˆ, t)
p ˆ , x ˆ
p ˆ ˆ x
x ˆ.. x ˆ ˆ x
x ...
ˆ x ˆ p ˆ
.
(5.32a)
x
k x
k x
x
0
! ˆ
0
! ˆ
k k
x
k k
x k times k times
Applying Eq. (25) k times to the last term in the parentheses, exactly as we did it in Eq. (26), we get
1 k
U
U
k
k
1
1
ˆ p , U ( ˆ x, t)
ik x
i
x
(5.32b)
x
ˆ
ˆ k 1
k
.
k
k x
k
k
k
1
! ˆ
1
(
)!
1
ˆ x
But the last sum is just the Taylor expansion of the derivative U/ x. Indeed, Chapter 5
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U
1
k'
k
U k
U k
k
'
1
' 1
'
1
U
ˆ x
ˆ x
x
(5.33)
k'
ˆ k 1
k
,
ˆ
x
k
k' x x
k
k x
k
k
k
'
0
! ˆ
ˆ
'0
'! ˆ ' 1
1
(
)!
1
ˆ x
where at the last step the summation index was changed from k’ to k – 1. As a result, we may rewrite Eq.
(5.32b) as
ˆ p , U(ˆ x, t)
,
(5.34)
x
i U(ˆ x, t)
ˆ x
so that Eq. (30) yields:
Heisenberg
d ˆ p
equation
x
U ( ˆ x, t).
(5.35)
for
dt
ˆ x
momentum
This equation also coincides with the classical equation of motion! Moreover, averaging Eqs. (29) and (35) over the initial state (as Eq. (4.191) prescribes), we get similar results for the expectation values:15
Ehrenfest
d x
p
d p
x
x
U
theorem
,
.
(5.36)
dt
m
dt
x
However, it is important to remember that the equivalence between these quantum-mechanical equations and similar equations of classical mechanics is superficial, and the degree of the similarity between the two mechanics very much depends on the problem. As one extreme, let us consider the case when a particle’s state, at any moment between t 0 and t, may be accurately represented by one, relatively px-narrow wave packet. Then we may interpret Eqs. (36) as the equations of the essentially classical motion of the wave packet’s center, in accordance with the correspondence principle. However, even in this case, it is important to remember the purely quantum mechanical effects of non-zero wave packet width and its spread in time, which were discussed in Sec. 2.2.
As an opposite extreme, let us revisit the “leaky” potential well discussed in Sec. 2.5 – see Fig.
2.15. Since both the potential U( x) and the initial wavefunction of that system are symmetric relative to point x = 0 at all times, the right-hand sides of both Eqs. (36) identically equal zero. Of course, the result they predict (that the average values of the coordinate and the momentum stay equal to zero at all times) is correct, but this fact does not tell us much about the rich dynamics of the system: the finite lifetime of the metastable state, the formation of two wave packets, their waveform and propagation speed (see Fig.
2.17), and about the insights the full solution gives for the quantum measurement theory and the system’s irreversibility. Another similar example is the energy band theory (Sec. 2.7), with its purely quantum effect of the allowed energy bands and forbidden energy gaps, of which Eqs. (36) give no clue.
To summarize, the Ehrenfest theorem is important as an illustration of the correspondence principle, but its predictive power should not be exaggerated.
5.3. The Feynman path integral
As has been already mentioned, even within the realm of wave mechanics, the bra-ket language may simplify some calculations that would be very bulky using the notation used in Chapters 1-3.
Probably the best example is the famous alternative, path-integral formulation of quantum mechanics.16
15 The equation set (36) constitutes the Ehrenfest theorem, named after its author, P. Ehrenfest.
16 This formulation was developed in 1948 by Richard Phillips Feynman. (According to his memories, this work was motivated by a “mysterious” remark by P. Dirac in his pioneering 1930 textbook on quantum mechanics.) Chapter 5
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I will review this important concept, cutting one math corner for the sake of brevity.17 (This shortcut will be clearly marked below.)
Let us inner-multiply both parts of Eq. (4.157a), which is essentially the definition of the time-evolution operator, by the bra-vector of state x,
x ( t) x ˆ u ( t, t ) ( t ) ,
(5.37)
0
0
insert the identity operator before the ket-vector on the right-hand side, and then use the closure condition in the form of Eq. (4.252), with x’ replaced with x 0:
x ( t) dx x ˆ u ( t, t ) x
x ( t ) .
(5.38)
0
0
0
0
0
According to Eq. (4.233), this equality may be represented as
( x, t)
dx x ˆ u( t, t ) x ( x , t ).
(5.39)
0
0
0
0
0
Comparing this expression with Eq. (2.44), we see that the long bracket in this relation is nothing other than the 1D propagator, which was discussed in Sec. 2.2, i.e.
G( x, t; x , t ) x ˆ u ( t, t ) x .
(5.40)
0
0
0
0
Let me hope that the reader sees that this equality corresponds to the physical sense of the propagator.
Now let us break the time segment [ t 0, t] into N (for the time being, not necessarily equal) parts, by inserting ( N – 1) intermediate points (Fig. 4) with
t t ... t ... t
t ,
(5.41)
0
1
k
N 1
and use the definition (4.157) of the time evolution operator to write
ˆ u ( t, t ) ˆ u ( t, t
) ˆ u ( t
, t
)...ˆ u ( t , t ) ˆ u ( t , t ) .
(5.42)
0
N 1
N 1
N 2
2
1
1
0
After plugging Eq. (42) into Eq. (40), let us insert the identity operator, again in the closure form (4.252), but written for xk rather than x’, between each two partial evolution operators including the time argument tk. The result is
G( x, t; x t ) dx
dx
... dx x ˆ u ( t, t
) x
x
ˆ u ( t
, t
) x
... x ˆ u ( t , t ) x . (5.43)
0, 0
N 1 N2 1
N 1
N 1
N 1
N 1
N 2
N 2
1
1
0
0
The physical sense of each integration variable xk is the wavefunction’s argument at time tk – see Fig. 4.
x
x
x
x
x
x
0
1
k
N 2
N 1
Fig. 5.4. Time partition and coordinate
notation at the initial stage of the
Feynman path integral’s derivation.
t
t
...
t
...
t
t
t
0
1
k
N 2
N 1
17 A more thorough discussion of the path-integral approach may be found in the famous text by R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, first published in 1965. (For its latest edition by Dover in 2010, the book was emended by D. Styler.) For a more recent monograph, which reviews more applications, see L. Schulman, Techniques and Applications of Path Integration, Wiley, 1981.
Chapter 5
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The key Feynman’s breakthrough was the realization that if all intervals are taken similar and sufficiently small, tk – tk- 1 = d → 0, all the partial brackets participating in Eq. (43) may be expressed via the free-particle’s propagator, given by Eq. (2.49), even if the particle is not free, but moves in a stationary potential profile U( x). To show that, let us use either Eq. (4.175) or Eq. (4.181), which, for a small time interval d, give the same result:
i
i ˆ 2
p
ˆ
ˆ
u (
d , ) exp
d
H exp
d U ˆ x
d
.
(5.44)
2
m
Generally, an exponent of a sum of two operators may be treated as that of c-number arguments, and in particular factored into a product of two exponents, only if the operators commute. (In this case, we can use all the standard algebra for the exponents of c-number arguments.) In our case, this is not so, because the operator p ˆ2 / 2 m does not commute with x ˆ , and hence with U( x ˆ ). However, it may be shown18 that for an infinitesimal time interval d, the non-zero commutator
ˆ 2
p
d , U ( ˆ x) d ,
0
(5.45)
2
m
proportional to ( d)2, may be ignored in the first, linear approximation in d. As a result, we may factorize the right-hand side in Eq. (44) by writing
i p ˆ2
i
u ˆ
( d , )
exp
exp
( ˆ
d 0
d
U x) d .
(5.46)
2 m
(This approximation is very much similar in spirit to the trapezoidal-rule approximation in the usual 1D
integration,19 which in also asymptotically impeachable.)
Since the second exponential function on the right-hand side of Eq. (46) commutes with the coordinate operator, we may move it out of each partial bracket participating in Eq. (43), with U( x) turning into a c-number function:
i ˆ 2
p
i
x
ˆ u( d , ) x x
exp
d x
U x
d
(5.47)
d
d
exp
( )
.
2
m
But the remaining bracket is just the propagator of a free particle, so that for it we may use Eq. (2.49):
1/ 2
i p ˆ2
m
m( dx)2
x exp
exp
.
(5.48)
d
d
x
i
2 m
2 i
d
2
d
As the result, the full propagator (43) takes the form
/ 2
m
N
N m( dx 2
)
U ( x)
G( x, t; x t ) lim
..
exp
. (5.49)
0, 0
d 0 dx
dx
dx
i
i
d
N 1
N 2
1
2 i d
k 1
2 d
N
18 This is exactly the corner I am going to cut because a strict mathematical proof of this (intuitively evident) statement would take more time/space than I can afford.
19 See, e.g., MA Eq. (5.2).
Chapter 5
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At N and hence d ( t – t 0)/ N 0, the sum under the exponent in this expression may be approximated with the corresponding integral:
N
i m dx 2
t
i m dx 2
U( x) d U( x) d,
(5.50)
k 1
2 d
2 d
t
t
0
k
and the expression in the square brackets is just the particle’s Lagrangian function L.20 The integral of this function over time is the classical action S calculated along a particular “path” x().21 As a result, defining the (1D) path integral as
N / 2
m
1D path
(...) [
D x( )] lim
dx
dx
.. dx (...),
(5.51a) integral:
d
0
N N
2
1
2
1
i
d
definition
N
we can bring our result to the following (superficially simple) form:
i
1D
G( x, t; x t ) exp S x D
x .
(5.51b) propagator
0, 0
( ) [ ( )]
via path
integral
The name “path integral” for the mathematical construct (51a) may be readily explained if we keep the number N of time intervals large but finite, and also approximate each of the enclosed integrals with a sum over M >> 1 discrete points along the coordinate axis – see Fig. 5a.
d
(a)
(b)
x
x
M
Fig. 5.5. Several 1D classical
x
x
paths: (a) in the discrete
0
0
approximation and (b) in the
t
t
t
continuous limit.
0
0
t
N 1
Then the path integral (51a) is the product of ( N – 1) sums corresponding to different values of time , each of them with M terms, each of those representing the function under the integral at a particular spatial point. Multiplying those ( N – 1) sums, we get a sum of ( N – 1) M terms, each evaluating the function at a specific spatial-temporal point [ x, ]. These terms may be now grouped to represent all possible different continuous classical paths x[] from the initial point [ x 0, t 0] to the finite point [ x, t]. It is evident that the last interpretation remains true even in the continuous limit N, M –
see Fig. 5b.
Why does such path representation of the sum make sense? This is because in the classical limit the particle follows just a certain path, corresponding to the minimum of the action S . As a result, for all close trajectories, the difference ( S – S cl) is proportional to the square of the deviation from the 20 See, e.g., CM Sec. 2.1.
21 See, e.g., CM Sec. 10.3.
