The Kronecker delta is usually de¯ned as De¯nition 22
±ij =
( 1 ifi=j,
0 ifi6=j. (1.61)
where i and j are integers.
However, the following equivalent de¯nition is found to be useful for the consideration of a continuous index analog of the Kronecker delta.
X
j
±ijfj =fi; (1.62)
where i and j are integers and fi represents thei-th member of an arbitrary sequence of
¯nite numbers.
The Dirac delta is an analog of the Kronecker delta with continuous indices. For continuous indices theiandj can be replaced by real numbersxandy and the Dirac delta is written as±(x¡y). The di®erence (x¡y) is used as the argument because the function can be seen to depend only on it. Likewise fi is replaced by a function f(x) of one real variable. f(x) must be ¯nite for allx. Hence, the continuous label analog of equation 1.62 produces the following de¯nition of the Dirac delta function.
De¯nition 23 Z
±(x¡y)f(y)dy=f(x); (1.63)
where f(x) is ¯nite everywhere. An integral with no limits shown explicitly is understood to have the limits ¡1to +1.
From this de¯nition it is seen that, f(x) being an arbitrary function, the only way equa-tion 1.63 is possible is if±(x¡y) is zero everywhere except at x= y. At x =y, ±(x¡y) would have to be in¯nite asdy is in¯nitesimal. Hence, the following are true for the Dirac delta function.
±(0) = 1; (1.64)
±(x) = 0 ifx6= 0: (1.65)
Because of the in¯nity in equation 1.64, the Dirac delta has meaning only when multiplied by a ¯nite function and integrated. Some identities involving the Dirac delta (in the same
CHAPTER 1. MATHEMATICAL PRELIMINARIES 14 integrated sense) that can be deduced from the de¯ning equation 1.63 are
±(x) = ±(¡x); (1.66)
The derivatives of a Dirac delta can be de¯ned once again in the sense of an integral. I shall consider only the ¯rst derivative±0(x).
Z
±0(x¡y)f(y)dy= ¡f(y)±(x¡y)j+¡11+ Z
±(x¡y)f0(y)dy; (1.73) where a prime denotes a derivative with respect to the whole argument of the function.
Thus Z
±0(x¡y)f(y)dy=f0(x): (1.74)
Some identities involving the±0(x) can be derived in the same fashion.
±0(x) = ¡±0(¡x); (1.75)
x±0(x) = ¡±(x): (1.76)
To understand the Dirac delta better it is very often written as the limit of some better known function. For example, has been de¯ned from the inner product. However, it is possible to ¯rst de¯ne the
CHAPTER 1. MATHEMATICAL PRELIMINARIES 15 norm and then the inner product as its consequence. Such an approach needs fewer rules but is more unwieldy. The inner product is then de¯ned as:
hrjsi = 1
2[jjri+jsij2¡ijjri+ijsij2 + (i¡1)(jjrij2+jjsij2)]:
Prove this result using the de¯nition of inner product and norm as given in this chapter.
2. In equation 1.8, show that a linearlydependentsetfjgiigwould give some of thejfii's to be the zero vector.
3. Using the de¯ning equation 1.11 of the commutators prove the identities in equa-tions 1.12 through 1.15.
4. Prove the following operator relations (for all operatorsPandQ,jsi 2 V, anda; b2 C) (a) (Qjsi)y=hsjQy
(b) Qyy =Q
(c) (aP +bQ)y=a¤Py+b¤Qy (d) (P Q)y=QyPy
(e) P Qis hermitian ifP and Q are hermitian and [P; Q] = 0.
(f) For a hermitian operatorH and jsi 2 V,hsjHjsi is real.
5. Prove the corollary 1.1.
6. Using the de¯ning equation 1.63 of the Dirac delta, prove the identities in equa-tions 1.66 through 1.72. For the derivative of the Dirac delta prove the identities in equations 1.75 and 1.76. [Hint: Remember that these identities have meaning only when multiplied by a ¯nite function and integrated.]
Chapter 2
The Laws (Postulates) of Quantum Mechanics
In the following, the term postulate will have its mathematical meaning i.e. an assumption used to build a theory. A law is a postulate that has been experimentally tested. All postulates introduced here have the status of laws.
2.1 A lesson from classical mechanics
There is a fundamental di®erence in the theoretical structures of classical and quantum mechanics. To understand this di®erence, one ¯rst needs to consider the structure of classical mechanics independent of the actual theory given by Newton. It is as follows.
1. The fundamental measured quantity (or the descriptor) of a sytem is its trajectory in con¯guration space (the space of all independent position coordinates describing the system). The con¯guration space has dimensionality equal to the number of degrees of freedom (sayn) of the system. So the trajectory is a curve inndimensional space parametrized by time. If xi is thei-th coordinate, then the trajectory is completely speci¯ed by the nfunctions of time xi(t). These functions are all observable.
2. A predictive theory of classical mechanics consists of equations that describe some initial value problem. These equations enable us to determine the complete trajectory xi(t) from data at some initial time. The Newtonian theory requires thexi and their time derivatives as initial data.
3. The xi(t) can then be used to determine other observables (sometimes conserved quantities) like energy, angular momentum etc.. Sometimes the equations of motion
16
CHAPTER 2. THE LAWS (POSTULATES) OF QUANTUM MECHANICS 17
can be used directly to ¯nd such quantities of interest.
The above structure is based on the nature of classical measurements. However, at small enough scales, such classical measurements (like the trajectory) are found to be experimentally meaningless. Thus, a di®erent theoretical structure becomes necessary. This structure is that of quantum mechanics. The structure of quantum mechanics, along with the associated postulates, will be stated in the following section. It is itemized to bring out the parallels with classical mechanics.
The reader must be warned that without prior experience in quantum physics the postulates presented here might seem rather ad hoc and \unphysical". But one must be reminded that in a ¯rst course in classical physics, Newton's laws of motion might seem just as ad hoc. Later, a short historical background will be given to partially correct this situation. About the \unphysical" nature of these postulates, very little can be done. Phe-nomena like the falling of objects due to gravity are considered \physical" due to our long term exposure to their repeated occurrence around us. In contrast, most of the direct evidence of quantum physics is found at a scale much smaller than everyday human experi-ence. This makes quantum phenomena inherently \unphysical". Hence, all one can expect of the following postulates is their self consistency and their ability to explain all observed phenomena within their range of applicabilty. To make quantum phenomena appear as
\physical" as classical phenomena, one needs to repeatedly experience quantum aspects of nature. Hence, this text (like most others) tries to provide as many examples as possible.
At ¯rst sight, the reader might also ¯nd the postulates to be too abstract and com-putationally intractable. The next two chapters should go a long way in correcting this problem.