216 24 Quantum Mechanics: Foundations24.3 The Canonical Commutation Relation In contrast to classical mechanics where observables correspond to arbitrary real functions f r, p of positi
Trang 1214 24 Quantum Mechanics: Foundations
However the l.h.s of (24.3) for ψ( r, t) does not necessarily agree pointwise
with the r.h.s for everyr, but the identity is only valid “almost everywhere”
in the following sense (so-called strong topology):
(|ψn → |ψ ⇐⇒
dV |ψn(r) − ψ(r)|2→ 0)
The coefficients c i and c(λ) are obtained by scalar multiplication from the
left with ψi| and ψλ|, i.e.,
In these equations the following orthonormalisation is assumed:
ψi|ψj = δi,j , ψλ |ψλ = δ(λ − λ) , ψi|ψλ = 0 , (24.5)
with the Kronecker delta δ i,j = 1 for i = j; δ i,j= 0 otherwise (i.e.,
j δ i,j f j=
f i for all complex vectors f i ), and the Dirac δ function δ(x), a so-called
generalized function (distribution), which is represented (together with the limit9 ε → 0) by a set {δε (x) }ε of increasingly narrow and at the same time increasingly high bell-shaped functions (e.g., Gaussians) with
∞
−∞
dxδ ε (x) ≡ 1 ),
defined in such a way that for all “test functions” f (λ) ∈ T (i.e., for all arbitrarily often differentiable complex functions f (λ), which decay for |λ| →
∞ faster than any power of 1/|λ|) one has the property (see Part II):
∞
−∞
dλδ(λ − λ) · f(λ) ≡ f(λ ), ∀f(λ) ∈ T (24.6)
This implies the following expression (also an extension of linear algebra!) for the scalar product of two vectors in Hilbert space after expansion in the basis belonging to an arbitrary observable ˆA (consisting of the orthonormal
proper and improper eigenvectors of ˆA):
1
ψ(1)ψ(2)2
i
c(1)i
∗
· c(2)
i +
dλ
c(1)(λ)
∗
· c(2)(λ) (24.7)
For simplicity it is assumed below, unless otherwise stated, that we are dealing with a pure point spectrum, such that in (24.7) only summations appear
9
The limit ε → 0 must be performed in front of the integral.
Trang 2a) However, there are important observables with a purely continuous
spec-trum (e.g., the position operator ˆ x with improper eigenfunctions
ψλ (x) := δ(x − λ) and the momentum operator ˆ pxwith improper eigenfunctions
ψλ (x) := (2π)−1/2 exp(iλ · x/) ; the eigenvalues appearing in (24.2) are then a(λ) = x(λ) = p(λ) = λ) b) In rare cases a third spectral contribution, the singular-continuous con-tribution, must be added, where it is necessary to replace the integral
3
dλ by a Stieltje’s integral
dg(λ) ,
with a continuous and monotonically nondecreasing, but nowhere
differen-tiable function g(λ) (the usual above-mentioned continuous contribution
is obtained in the differentiable case g(λ) ≡ λ).
For a pure point spectrum one can thus use Dirac’s abstract bra-ket formalism: a) “observables” are represented by self-adjoint10operators In diagonal rep-resentation they are of the form
ˆ
i
with real eigenvalues a i and orthonormalized eigenstates|ψi,
ψi|ψk = δik ,
and
b) the following statement is true (which is equivalent to the expansion the-orem (24.5)):
ˆ
1 =
i
Equation (24.9) is a so-called “resolution of the identity operator ˆ1” by
a sum of projections
ˆ
Pi :=|ψiψi|
The action of these projection operators is simple:
ˆ
Pi|ψ = |ψi ψi|ψ = |ψici
Equation (24.9) is often applied as
|ψ ≡ ˆ1|ψ
10The difference between hermiticity and self-adjointness is subtle, e.g instead
of vanishing values at the boundaries of an interval one only demands periodic behavior In the first case the differential expression for ˆp xhas no eigenfunctions for the interval, in the latter case it has a complete set
Trang 3216 24 Quantum Mechanics: Foundations
24.3 The Canonical Commutation Relation
In contrast to classical mechanics (where observables correspond to arbitrary real functions f ( r, p) of position and momentum, and where for products
of x i and p j the sequential order does not matter), in quantum mechanics
two self-adjoint operators representing observables typically do not commute.
Instead, the following so-called canonical commutation relation holds11:
[ˆp j , ˆ x k] := ˆp j xˆk − ˆxk pˆj ≡
The canonical commutation relation does not depend on the representa-tion (see below) It can be derived in the wave mechanics representarepresenta-tion by
applying (24.10) to an arbitrary function ψ( r) (belonging of course to the
maximal intersectionImax of the regions of definition of the relevant opera-tors) Using the product rule for differentiation one obtains
i {∂(xk ψ( r))/∂x j − xk∂ψ( r)/∂x j} ≡
iδjkψ( r) , ∀ψ(r) ∈ I max ⊆ HR
(24.11) This result is identical to (24.10)
24.4 The Schr¨ odinger Equation; Gauge Transformations
Schr¨odinger’s equation describes the time-development of the wave function
ψ( r, t) between two measurements This fundamental equation is
−
i
∂ψ
ˆ
H is the so-called Hamilton operator of the system, see below, which
corre-sponds to the classical Hamiltonian, insofar as there is a relationship between classical and quantum mechanics (which is not always the case12) This
im-portant self-adjoint operator determines the dynamics of the system Omitting spin (see below) one can obtain the Hamilton operator directly from the Hamilton function (Hamiltonian) of classical mechanics by replacing
the classical quantities r and p by the corresponding operators, e.g.,
ˆ
x |ψ → x · ψ(r) ; ˆpx|ψ →
i
∂
∂x ψ( r)
11 Later we will see that this commutation relation is the basis for many important relations in quantum mechanics
12 For example the spin of an electron (see below) has no correspondence in classical mechanics
Trang 4For example, the classical Hamiltonian H( r, p, t) determining the motion
of a particle of mass m and electric charge e in a conservative force field due to
a potential energy V ( r) plus electromagnetic fields E(r, t) and B(r, t), with
corresponding scalar electromagnetic potential Φ( r, t) and vector potential A(r, t), i.e., with
B = curlA and E = −gradΦ − ∂ A
∂t ,
is given by
H( r, p, t) = (p − eA(r, t))2
2m + V ( r) + eΦ(r, t) (24.13) The corresponding Schr¨odinger equation is then
−
i
∂ψ( r, t)
∂t = ˆHψ =
1
2m
i∇ − e · A(r, t)
2
ψ( r, t)
+{V (r) + e · Φ(r, t)} ψ(r, t) (24.14) The corresponding Newtonian equation of motion is the equation for the Lorentz force (see Parts I and II)13
mdv
dt =−∇V (r) + e · (E + v × B) (24.15)
Later we come back to these equations in connection with spin and with the Aharonov-Bohm effect.
In (24.13) and (24.14), one usually setsA ≡ 0, if B vanishes everywhere.
But this is neither necessary in electrodynamics nor in quantum mechanics
In fact, by analogy to electrodynamics, see Part II, a slightly more complex
gauge transformation can be defined in quantum mechanics, by which certain
“nonphysical” (i.e., unmeasurable) functions, e.g., the probability amplitude
ψ( r, t), are non-trivially transformed without changes in measurable
quanti-ties
To achieve this it is only necessary to perform the following simultaneous changes ofA, Φ and ψ into the corresponding primed quantities14
A (r, t) = A(r, t) + ∇f(r, t) (24.16)
Φ (r, t) = Φ(r, t) − ∂f ( r, t)
ψ (r, t) = exp
,
+ie · f ( r, t)
13With the HamiltonianH =(p−eA)2
2m one evaluates the so-called canonical
equa-tions ˙ x = ∂ H/∂p x, ˙p x=−∂H/∂x, where it is useful to distinguish the canonical momentum p from the kinetic momentum mv := p − eA.
14
Here we remind ourselves that in classical mechanics the canonical momentum
p must be gauged In Schr¨odinger’s wave mechanics one has instead
i∇ψ, and
only ψ must be gauged.
Trang 5218 24 Quantum Mechanics: Foundations
with an arbitrary real function f ( r, t) Although this transformation changes
both the Hamiltonian ˆH, see (24.14), and the probability amplitude ψ( r, t),
all measurable physical quantities, e.g., the electromagnetic fields E and B
and the probability density|ψ(r, t)|2as well as the probability-current density
(see below, (25.12)) do not change, as can be shown.
24.5 Measurement Process
We shall now consider a general state |ψ In the basis belonging to the
observable ˆA (i.e., the basis is formed by the complete set of orthonormalized eigenvectors |ψi and |ψλ of the self-adjoint ( ˆ= hermitian plus complete) operator ˆ A), this state has complex expansion coefficients
ci =ψi|ψ and c(λ) = ψλ|ψ
(obeying the δ-conditions (24.5)) If in such a state measurements of the
observable ˆA are performed, then
a) only the values a i and a(λ) are obtained as the result of a single
measure-ment, and
b) for the probability W ( ˆ A, ψ, Δa) of finding a result in the interval
Δa := [amin, amax) ,
the following expression is obtained:
W ( ˆ A, ψ, Δa) =
a i ∈Δa
|ci|2+
a(λ) ∈Δa
dλ |c(λ|2 . (24.19)
In this way we obtain what is known as the quantum mechanical expectation value, which is equivalent to a fundamental experimental value, viz the
aver-age over an infinitely long series of measurements of the observable ˆA in the state ψ:
(A) ψ
(i)
i ai|ci|2+
dλa(λ) |c(λ)| 2 (ii)= ψ| ˆ A |ψ (24.20)
Analogously one finds that the operator
δ ˆ A
2
:=
ˆ
A − (A)ψ2 corresponds to the variance (the square of the mean variation) of the values
of a series of measurements around the average
Trang 6For the product of the variances of two series of measurements of the observ-ables ˆA and ˆ B we have, with the commutator
[ˆa, ˆ b] := ˆ A ˆ B − ˆ B ˆ A , Heisenberg’s uncertainty principle:
ψ|δ ˆ A
2
|ψ · ψ|δ ˆ B
2
|ψ ≥ 1
4
ψ|A, ˆˆ B
|ψ2
Note that this relation makes a very precise statement; however one should
also note that it does not deal with single measurements but with expectation
values, which depend, moreover, on|ψ.
Special cases of this important relation (which is not hard to derive) are obtained for
ˆ
A = ˆ px , B = ˆˆ x with
ˆ
A, ˆ B
=
i , and for the orbital angular moments15
ˆ
A = ˆ L x and B = ˆˆ L y with
ˆ
A, ˆ B
= iˆL z
On the other hand permutable operators have identical sets of eigenvectors (but different eigenvalues)
Thus in quantum mechanics a measurement generally has a finite influ-ence on the state (e.g., |ψ → |ψ1; state reduction), and two series of mea-surements for the same state |ψ, but non-commutable observables ˆ A and ˆ B, typically (this depends on |ψ!) cannot simultaneously have vanishing expec-tation values of the variances16.
24.6 Wave-particle Duality
In quantum mechanics this important topic means that
a) (on the one hand) the complex probability amplitudes (and not the prob-abilities themselves) are linearly superposed, in the same way as field
amplitudes (not intensities) are superposed in coherent optics, such that
interference is possible (i.e., |ψ1+ ψ2|2 =|ψ1|2+|ψ2|2+ 2Re(ψ ∗
1· ψ2)); whereas
15
and also for the spin momenta (see below)
16Experimentalists often prefer the “short version” “ cannot be measured
si-multaneously (with precise results)”; unfortunately this “shortening” gives rise
to many misunderstandings In this context the relevant section in the “Feyn-man lectures” is recommended, where it is demonstrated by construction that for a single measurement (but of course not on average) even ˆp x and ˆx can
simultaneously have precise values
Trang 7220 24 Quantum Mechanics: Foundations
b) (on the other hand) measurement and interaction processes take place with single particles such as photons, for which the usual fundamental
conservation laws (conservation of energy and/or momentum and/or an-gular momentum) apply per single event, and not only on average (One
should mention that this important statement had been proved experi-mentally even in the early years of quantum mechanics!)
For example, photons are the “particles” of the electromagnetic wave field, which is described by Maxwell’s equations They are realistic objects, i.e (massless) relativistic particles with energy
E = ω and momentum p = k
Similarly, (nonrelativistic) electrons are the quanta of a “Schr¨odinger field”17, i.e., a “matter field”, where for the matter field the Schr¨odinger equation plays the role of the Maxwell equations
The solution of the apparent paradox of wave-particle duality in quan-tum mechanics can thus be found in the probabilistic interpretation of the wave function ψ This is the so-called “Copenhagen interpretation” of quan-tum mechanics, which dates back to Niels Bohr (in Copenhagen) and Max Born (in G¨ ottingen) This interpretation has proved to be correct without contradiction, from Schr¨ odinger’s discovery until now – although initially the interpretation was not undisputed, as we shall see in the next section.
24.7 Schr¨ odinger’s Cat: Dead and Alive?
Remarkably Schr¨odinger himself fought unsuccessfully against the Copen-hagen interpretation 18 of quantum mechanics, with a question, which we
paraphrase as follows: “What is the ‘state’ of an unobserved cat confined in
a box, which contains a device that with a certain probability kills it immedi-ately? ”
Quantum mechanics tends to the simple answer that the cat is either in
an “alive” state (|ψ = |ψ1), a “dead” state(|ψ = |ψ2) or in a (coherently)
“superposed” state (|ψ ≡ c1|ψ1 + c2|ψ2).
With his question Schr¨odinger was in fact mainly casting doubt on the idea that a system could be in a state of coherent quantum mechanical
super-position with nontrivial probabilities of states that are classically mutually
17 Relativistic electrons would be the quanta of a “Dirac field”, i.e., a matter field,
where the Dirac equation, which is not described in this book, plays the role of
the Maxwell equations of the theory
18 Schr¨odinger preferred a “charge-density interpretation” of e|ψ(r)|2 But this
would have necessitated an addition δ V to the potential energy, i.e., classically:
δ V ≡ e2RR
dV dV |ψ(r)|2|ψ(r )|2/(8πε0|r−r |), which – by the way – after a
sys-tematic quantization of the corresponding classical field theory (the so-called
“2nd quantization”) leads back to the usual quantum mechanical single-particle Schr¨odinger equation without such an addition, see [24].
Trang 8exclusive (“alive” and “dead” simultaneously!) Such states are nowadays
called “Schr¨odinger cat states”, and although Schr¨odinger’s objections were erroneous, the question led to a number of important insights For
exam-ple, in practice the necessary coherence is almost always destroyed if one deals with a macroscopic system This gives rise to corresponding quantita-tive terms such as the coherence length and coherence time In fact there are many other less spectacular “cat states”, e.g the state describing an object
which is simultaneously in the vicinity of two different places,
ψ = c1ψ x ≈x1+ c2ψ x ≈x2.
Nowadays one might update the question for contemporary purposes For example we could assume that Schr¨odinger’s proverbial cat carries a bomb attached to its collar19, which would not only explode spontaneously with
a certain probability, but also with certainty due to any external interaction
process (“measurement with interaction”) The serious question then arises
as to whether or not it would be possible to verify by means of a “quantum measurement without interaction”, i.e., without making the bomb explode, that a suspicious box is empty or not
This question is treated below in Sect 36.5; the answer to this question is
actually positive, i.e., there is a possibility of performing an “interaction-free
quantum measurement”, but the probability for an interaction (⇒ explosion),
although reduced considerably, does not vanish completely For details one should refer to the above-mentioned section or to papers such as [31]
19If there are any cat lovers reading this text, we apologize for this thought exper-iment
Trang 925 One-dimensional Problems
in Quantum Mechanics
In the following we shall deal with stationary states For these states one can make the ansatz :
ψ( r, t) = u(r) · e −i Et
.
As a consequence, for stationary states, the expectation values of a constant observable ˆA is also constant (w.r.t time):
ψ(t)| ˆ A |ψ(t) ≡ u| ˆ A |u The related differential equation for the amplitude function u( r) is called
the time-independent Schr¨ odinger equation In one dimension it simplifies for vanishing electromagnetic potentials, Φ = A ≡ 0, to:
u =−k2(x)u(x) , with k2(x) := 2m
2 (E − V (x)) (25.1) This form is useful for values of
x where E > V (x) , i.e., for k2(x) > 0
If this not the case, then it is more appropriate to write (25.1) as follows
u = +κ2(x)u(x) , with κ2(x) := 2m
2(V (x) − E) (25.2) For a potential energy that is constant w.r.t time, we have the following general solution:
u(x) = A+exp(ik · x) + A − · exp(−ik · x) and
u(x) = B+exp(+κ · x) + B− · exp(−κ · x)
Using
cos(x) := (exp(ix) + exp( −ix))/2 and sin(x) := (exp(ix) − exp(−ix))/(2i)
Trang 10we obtain in the first case
u(x) = C+· cos(κ · x) + C− · sin(κ) , where
C+= A++ A − and
C − = i(A+− A− ) The coefficients A+, A − etc are real or complex numbers, which can be determined by consideration of the boundary conditions
25.1 Bound Systems in a Box (Quantum Well); Parity
Assume that
V (x) = 0 for |x| ≥ a , whereas
V (x) = −V0(< 0) for |x| < a The potential is thus an even function,
V (x) ≡ V (−x) , ∀x ∈ R ,
cf Fig 25.1 Therefore the corresponding parity is a “good quantum number”
(see below)
Fig 25.1 A “quantum well” potential and a sketch of the two lowest
eigenfunc-tions A symmetrical quantum-well potential of width Δx = 2 and depth V0 = 1
is shown as a function of x In the two upper curves the qualitative behavior of
the lowest and 2nd-lowest stationary wave functions (→ even and odd parity) is
sketched, the uppermost curve with an offset of 0.5 units Note that the quantum
mechanical wave function has exponential tails in the external region which a clas-sical bound particle never enters In fact, where the clasclas-sical bound particle has
a point of return to the center of the well, the quantum mechanical wave function only has a turning point, i.e., only the curvature changes sign