16 1. The Magic of Quantum Mechanics
moving along a single coordinate axis x(the mathematical foundations of quantum mechanics are given in Appendix B on p. 895).
Postulate I (on the quantum mechanical state)
wave function
The state of the system is described by the wave function 9=9(x;t), which depends on the coordinate of particle x at time t. Wave functions in general are complex functions of real variables. The symbol 9∗(x;t) denotes the complex conjugate of 9(x;t). The quantity
p(x;t)=9∗(x;t)9(x;t)dx=¯9(x;t)¯2 dx (1.1)
gives the probability that at time t the x coordinate of the particle lies in the small interval [x;x+dx] (Fig. 1.3.a). The probability of the particle being in the interval (a;b) on the x axis is given by (Fig. 1.3.b): a |9(x;t)|2 dx.
statistical interpretation
probability density
normalization
The probabilistic interpretation of the wave function was proposed by Max Born.26 By analogy with the formula: mass = density × volume, the quantity 9∗(x;t)9(x;t) is called the probability density that a particle at time t has posi-tion x.
In order to treat the quantity p(x;t) as a probability, at any instant t the wave function must satisfy the normalization condition:
Z ∞
9 (x;t)9(x;t)dx=1: (1.2) −∞
probability probability
Fig. 1.3. A particle moves along the x axis and is in the state described by the wave function 9(x;t). Fig. (a) shows how the probability of ﬁnding particle in an inﬁnitesimally small section of the length dx
at x0 (at time t =t0) is calculated. Fig. (b) shows how to calculate the probability of ﬁnding the particle at t =t0 in a section (a;b).
26M. Born, Zeitschrift für Physik 37 (1926) 863.
1.2 Postulates 17
All this may be generalized for more complex situations. For example, in three-dimensional space, the wave function of a single particle depends on position r = (x;y;z) and time: 9(r;t), and the normalization condition takes the form
Z ∞ Z ∞ Z ∞ Z
dx dy dz9 (x;y;z;t)9(x;y;z;t)≡ 9 (r;t)9(r;t)dV −∞ −∞ −∞
Z
≡ 9∗(r;t)9(r;t)d3r =1: (1.3)
When integrating over whole space, for simplicity, the last two integrals are given without the integration limits, but they are there implicitly, and this convention will be used by us throughout the book unless stated otherwise.
For n particles (Fig. 1.4), shown by vectors r1;r2;:::;rn in three-dimensional space, the interpretation of the wave function is as follows. The probability P, that at a given time t =t0, particle 1 is in the domain V1, particle 2 is in the domain V2 etc., is calculated as
Z Z Z
P = dV1 dV2 ::: dVn 9∗(r1;r2;:::;rn;t0)9(r1;r2;:::;rn;t0) V1 V2 Vn
Z Z Z
≡ d3r1 d3r2 ::: d3rn 9∗(r1;r2;:::;rn;t0)9(r1;r2;:::;rn;t0): V1 V2 Vn
Ofteninthisbookwewillperformwhatiscalledthenormalizationofafunction, normalization which is required if a probability is to be calculated. Suppose we have a unnormal-
Fig. 1.4. Interpretation of a many-particle
wave function, an example for two particles. The number |ψ(r1;r2;t0)|2 dV1 dV2 repre-sents the probability that at t = t0 particle 1 is in its box of volume dV1 shown by vec-tor r1 and particle 2 in its box of volume dV2 indicated by vector r2.
18 1. The Magic of Quantum Mechanics
ized function27 ψ, that is
Z ∞
ψ(x;t) ψ(x;t)dx=A; (1.4) −∞
with 0 < A = 1. To compute the probability ψ must be normalized, i.e. multiplied by a normalization constant N, such that the new function 9 = Nψ satisﬁes the normalization condition:
1 =Z ∞ 9∗(x;t)9(x;t)dx= N∗NZ ∞ ψ∗(x;t)ψ(x;t)dx=A|N|2: −∞ −∞
Hence, |N|= √A. How is N calculated? One person may choose it as equal to N = √ ,another: N =−√ ,athird: N =e1989i √ ,andsoon.Thereare,therefore,an
phase inﬁnite number of legitimate choices of the phase φ of the wave function 9(x;t)= eiφ 1 ψ. Yet, when 9∗(x;t)9(x;t); is calculated, everyone will obtain the same
result, 1 ψ∗ψ, because the phase disappears. In most applications, this is what will happen and therefore the computed physical properties will not depend on the choice of phase. There are cases, however, where the phase will be of importance.
Postulate II (on operator representation of mechanical quantities)
The mechanical quantities that describe the particle (energy, the compo-nents of vectors of position, momentum, angular momentum, etc.) are rep-resentedbylinearoperatorsactinginHilbertspace(seeAppendixB).There are two important examples of the operators: the operator of the particle’s position x = x (i.e. multiplication by x, or x = x·, Fig. 1.5), as well as the operator of the (x-component) momentum px =−ih d , where i stands for the imaginary unit.
Note that the mathematical form of the operators is always deﬁned with respect to a Cartesian coordinate system.28 From the given operators (Fig. 1.5) the oper-ators of some other quantities may be constructed. The potential energy operator
V =V (x), where V (x) [the multiplication operator by the function V f =V (x)f] represents afunctionof xcalleda potential.The kineticenergy operatorofasingle
particle (in one dimension) T = 2m =−2m dx2 , and in three dimensions:
T = 2m = ˆx2 + 2m +pz2 =−2m1; (1.5)
27Eq. (1.2) not satisﬁed.
28Although they may then be transformed to other coordinates systems.
1.2 Postulates 19
Mechanical quantity
coordinate
momentum component
kinetic energy
Classical formula
x
px
T = 22 = 2m
Operator acting on f
xf def xf
pxf = −ih∂f
Tf =−2m1f
Fig. 1.5. Mechanical quantities and the corresponding operators.
where the Laplacian 1 is
2 2 2
1≡ ∂x2 + ∂y2 + ∂z2 (1.6)
and m denotes the particle’s mass. The total energy operator, or Hamiltonian is
the most frequently used: Hamiltonian
H =T +V : (1.7)
An important feature of operators is that they may not commute,29 i.e. for two particular operators A and B one may have AB −BA = 0. This property has im-portant physical consequences (see below, postulate IV and the Heisenberg uncer-tainty principle). Because of the possible non-commutation of the operators, trans-formation of the classical formula (in which the commutation or non-commutation did not matter) may be non-unique. In such a case, from all the possibilities one has to choose an operator which is Hermitian. The operator A is Hermitian if, for
any functions ψ and φ from its domain, one has commutation
Z ∞ Z ∞
ψ (x)Aφ(x)dx= [Aψ(x)] φ(x)dx: (1.8) −∞ −∞
Using what is known as Dirac notation, Fig. 1.6, the above equality may be written in a concise form:
hψ|Aφi=hAψ|φi: (1.9)
In Dirac notation30 (Fig. 1.6) the key role is played by vectors bra: h | and ket: bra and ket | i denoting respectively ψ∗ ≡ hψ| and φ ≡ |φi. Writing the bra and ket as hψ||φi
29Commutation means AB =BA.
30Its deeper meaning is discussed in many textbooks of quantum mechanics, e.g., A. Messiah, “Quan-tum Mechanics”, vol. I, Amsterdam (1961), p. 245. Here we treat it as a convenient tool.
20
Z
ψ∗φdτ ≡hψ|φi
Z
ψ∗Aφdτ ≡hψ|Aφi
or hψ|A|φi
Q=|ψihψ|
X
1 = |ψkihψk|
k
1. The Magic of Quantum Mechanics
Scalar product of two functions
Scalar product of ψ and Aφ or
a matrix element of the operator A
Projection operator on the direction of the vector ψ
Spectral resolution of identity. Its sense is best seen when acting on χ:
χ= k |ψkihψk|χi= k |ψkick.
Fig. 1.6. Dirac notation.
denotes hψ|φi, or the scalar product of ψ and φ in a unitary space (Appendix B), while writing it as |ψihφ| means the operator Q =|ψihφ|, because of its action on function ξ =|ξi shown as: Qξ =|ψihφ|ξ =|ψihφ|ξi=cψ, where c =hφ|ξi.
The last formula in Fig. 1.6 (with {ψk} standing for the complete set of func-tions) represents what is known as “spectral resolution of identity”, best demon-strated when acting on an arbitrary function χ:
X X
χ= |ψkihψk|χi= |ψkick:
k k
spectral resultion of identity
time evolution
We have obtained the decomposition of the function (i.e. a vector of the Hilbert space) χ on its components |ψkick along the basis vectors |ψki of the Hilbert space. The coefﬁcient ck = hψk|χi is the corresponding scalar product, the basis vectors ψk are normalized. This formula says something trivial: any vector can be retrieved when adding all its components together.
Postulate III (on time evolution of the state)
TIME-DEPENDENT SCHRÖDINGER EQUATION
The time-evolution of the wave function 9 is given by the equation
ih∂9(x;t) =H9(x;t); (1.10)
where H is the system Hamiltonian, see eq. (1.7).
H may be time-dependent (energy changes in time, interacting system) or time-independent (energy conserved, isolated system). Eq. (1.10) is called the time-dependent Schrödinger equation (Fig. 1.7).
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