The measurement postulate (postulate 4) relates the quantum theoretic predictions with experimental observations. Those predictions are found to be in conformity with experimental observations. That postulate, however, has been the subject of debate right since its conception. The issues debated are: (i) whether the measurement postulate is consistent with the postulate specifying the time evolution (postulate 5) and (ii) whether it denies the objective reality per the criterion of the Einstein, Podolsky and Rosen (EPR). The said issues are highlighted in the form of paradoxes by the thought experiment (Gedanken experiment in German) of Schrödinger, and that of EPR. See [40] for collection of articles and papers on the problem of measurement. See in particular [42] and [43].

The Measurement Problem

We recall first the measurement postulate. Let a system S (hereafter called the object) be subject to the measurement of an observable . Let the eigenvalues of be non-degenerate with as the corresponding orthonormal eigenvectors. Let the object be in the state jsi before the measurement, where

According to the measurement postulate, the measurement of would make the state of the object collapse to one of the eigenstates of and the outcome of the measurement would be the eigenvalue corresponding to the eigenstate to which the state of the object has collapsed. Hence, if the outcome of the measurement is the eigenvalue kk then we know that the state of the object after the measurement is, written symbolically as.

Not many problems in quantum mechanics are analytically exactly solvable. The methods of solving the equations approximately therefore play an important role. Though construction of approximate solution would depend on a particular problem, there are methods, discussed in this chapter, which are of general interest as they are applicable to widely occurring situations. These methods can be divided into two categories: one for time-independent and the other for time-dependent Hamiltonians. These categories can be divided further into two classes: one in which the Hamiltonian can be divided in to a strong and a weak part allowing the weak part to be treated as a perturbation, and the other in which certain gross features of the interaction suggest approximation methods for its treatment. The quasi-classical approximation for stationary states, and the adiabatic and sudden approximations for time-dependent Hamiltonians are the examples of non-perturbative techniques.

WKB Approximation

The quantum theory is expected to approach the classical theory in the limit of de Broglie wavelength becoming small compared with the distances over which the potential varies. In that limit, the Schrödinger equation reduces to the Hamilton–Jacobi equation of classical mechanics. The method outlined in this section consists in finding that solution of the Schrödinger equation which contains lowest order quantum effects when the conditions for the classical approximation hold. We will see that this approximation closely reproduces the quantization rule of the old quantum theory. Consider the time-independent Schrödinger equation of a particle of mass m in the time-independent potential V(r),

This chapter builds the formal structure of quantum mechanics on the basis of five postulates. These postulates unify and formalize the concepts of matrix and wave mechanics introduced in Chapter 1. The first three postulates provide the framework for mathematical modelling of isolated systems and observables. The fourth postulate establishes relationship between theoretical description and experimental observations. Describing time evolution of a system is the content of the fifth postulate. The formalism is then extended to describe a composite system in terms of its constituent subsystems.

The formalism for describing a subsystem of a composite system, leading to the concept of density operator, is developed in Chapter 6.

Postulate 1: On Representing an Isolated System

An isolated system is described by a vector in the Hilbert space. The vectors differing only by a multiplying constant represent the same physical state.

A vector representing the state of a system is also referred to as a state vector. The implications of this postulate are

In Chapter 2, we showed that the vectors in the Hilbert space can be represented by column vectors and the operators by n × n matrices. The algebra of operators in a finite dimensional space is thus equivalent with the algebra of finite dimensional matrices. In this chapter we summarize some results of relevance to us in matrix algebra.

Matrix Algebra

Consider the space of finite dimension n spanned by the orthonormal basis f﹛ |k﹜. As discussed in Chapter 2, a vector jui in the space is represented by the column vectorue constituted by as its elements, whereas its dual huj is represented

†. Similarly an operator AO is represented by the n × n matrix AQ constituted by as its elements. We restate below in matrix notation some notions of abstract operator algebra introduced in Chapter 2.

1. In terms of their representation by column vectors, the scalar or inner product hujvi between jui and is given by

where ﹛ṵ﹜ and ﹛ṵ﹜ are the elements ofṵ and ṵ , respectively. It may be verified that the definition of the scalar product given above is in accordance with the axioms of the scalar product.

Let ﹛ui﹜ be an orthonormal basis spanning the given space. By virtue of (3.1) and (3.2), the completeness and the orthonormality relations,

This chapter solves the one-dimensional Schrödinger equation in a potential which (i) is constant everywhere or (ii) jumps discontinuously from one constant value to another at a finite number of points. The importance of studying such simple potentials lies in the fact that the motion of a particle in them exhibits certain exclusive quantum features, such as the possibility of finding the particle in the classically forbidden region or tunnelling through it. They also idealize several realistic potentials and provide useful insight in to the properties of the motion of a particle in realistic situations.

Constant Potential

Consider first the potential which has the same value on the entire real axis. A particle moving in such a potential would not experience any force and hence propagate freely. Without loss of generality, we take the potential to be zero. If the mass of the particle is m, then its wave function of definite energy E is the solutions of the Schroödinger equation (9.1) corresponding to U(x) = 0:

Recall from Section 8.6 that the energy of the particle cannot be less than the global minimum of the potential in which it moves. Since the potential in the present case is zero everywhere, we must have. The two linearly independent solutions of the equation above then read

This Appendix addresses the question of solving a linear ordinary second-order homogeneous differential equation

We do not go in to the conditions on the functions P(x) and Q(x) required for the equation above to admit a solution but assume that the functions possess the desired properties. The explicit solution will depend, of course, on the functional forms of P(x) and Q(x). However, before finding explicit solutions, we list below some properties of the solutions of (B.1) which are independent of the functional forms of P(x) and Q(x).

• 1. If y1(x) and y2(x) are two solutions of (B.1) then on substituting in it y(x) = ay1(x) + βy2(x), where a and b are complex numbers, it may be seen that y(x) also satisfies that equation.

• 2. Since, as shown above, any linear combination of the solutions of (B.1) is also a solution, it is sufficient to find all its linearly independent solutions. Any other solution can then be expressed as a linear combination of those linearly independent ones. To ascertain whether the solutions y1(x), y2(x) are linearly independent, assume that there exist constants A and B such that

The functions y1(x), y2(x) are linearly independent if the equation above is solved only by A = B = 0. The constants A, B are determined by forming second equation by differentiating (B.2) and solving the equation so obtained simultaneously with (B.2) by writing them as

In Chapter 15, we showed that invariance under rotation in real three-dimensional position space transforms the state of a system by a unitary transformation generated by the operator which is the sum of the orbital angular momentum operator, and the spin operator which acts on the internal state of the system. The components of obey the same commutation relations as the corresponding components of. The eigenvalue problem of the orbital angular momentum has been solved in Chapter 14 by working in the position representation. We found that the eigenvalues of are

where l = 0, 1, 2,….

However, acts on the internal state of the system and does not have position representative. Hence, for the systems having spin, we need to solve the eigenvalue problem of the angular momentum by invoking only the commutation relations between the components of, the task undertaken in this chapter. We will find that, like the orbital angular momentum, the eigenvalues of are also expressible in the form but, in addition to non-negative integral values, can be a half-odd positive integer. Thus, the spectrum of derived using the commutation relations turns out to be different from that obtained while working in position representation when permissible. This is unlike the case of harmonic oscillator for which the eigenvalues turn out to be same whether the oscillator problem is solved in the position representation (when permissible) or by using only the commutation relations between the harmonic oscillator operators.

The concept of Hilbert space is at the heart of the body of quantum mechanics. In this chapter, we introduce the concept of Hilbert space and develop the algebra and calculus of operators in it. It is preceded by introduction of the notions of ‘linear vector space’ and ‘scalar product space’ on which the concept of Hilbert space is based.

Vector Space

A set V of elements {|u, |vi, |w,…g, called vectors, is said to constitute a vector space V over the field F if it is closed under the operations of addition, and multiplication by scalars in F governed by following axioms:

The symbol ji, introduced by Dirac to denote a vector, is called a ket. In the study of the representation of the vector space, we will see that a vector can be represented in different ways. A representation of interest is the one in which it is represented by a column vector. We denote column vectors by a tilde under a letter. Thus, for example,ṵ ewould represent a column vector.

Because of its properties being similar to those of the scalar zero, the null vector j0i may be denoted by the scalar 0. Whether 0 in an expression involving vectors refers to the null vector j0i or to the scalar 0 is to be inferred from the context.

The vector spaces of interest to us are:

1. If the field F in the axioms defining the vector space V is the field of real numbers, then it is called the vector space over the field of real numbers or simply a real vector space.

Entanglement characterizes such correlations between the subsystems of a system which possess exclusive quantum features, in the sense to be described in this chapter. The concept of exclusive quantum features, often called quantumness, stems from the local hidden variables (LHV) theory. We describe in particular the approaches to identify the quantumness of correlations between two spin-1/2 particles. Of various such approaches, the one based on the concept of joint quasiprobability (JQP) distribution provides necessary and sufficient condition for the said identification. The quantumness of correlations between spin-1/2 particles is a resource in quantum computation and quantum information.

Entanglement

We discuss the concept of only the bipartite entanglement, i.e., entanglement between two subsystems of a system. Let the subsystems be named A and B. Let the state of the combined system A + B be described by the density operator. Let and be the operators acting only on A and only on B, respectively. If

for all and then A and B are said to be uncorrelated. The equation above will hold if

where is the density matrix of A alone and that for B alone. For a pure state of the combined system, the condition (18.2) for (18.1) to hold is equivalent with the condition

This chapter discusses the problem of solving the Schrödinger equation for the potentials which are central: a potential V(r) is called central if it depends on the magnitude jrj of r and not on its orientation. We discuss, in particular, the problems of a free particle, potential well, hydrogen atom and of an isotropic harmonic oscillator.

In the case of central potentials, it is convenient and instructive to work in the spherical polar coordinate system discussed next.

Spherical Polar Coordinates

Let (x, y, z) be the coordinates of a point P with respect to the Cartesian system centred at O with ex, ey, ez as the unit vectors along the three axes.

The spherical polar system is described in terms of the coordinates (r, h, ɸ) where, as shown in Figure 14.1, r = jrj is the magnitude of the position vector of P, θ is the angle between OP and the z-axis, and _ is the angle between the x-axis and the projection of OP on the x − y plane. The spherical polar coordinates of P are related to its Cartesian coordinates by the relations

In spherical coordinate description, a potential is central if it is a function only of r. We list below some results useful for the discussion to follow.

A problem of considerable interest is the theory of motion of electrons in a bulk or in a nano sized crystalline solid. Whereas the potential in a bulk solid may be considered to be periodic in entire space, that in a nano sized solid is periodic in a finite region of space. In this chapter, we develop the formalism to study the motion of a particle in a one-dimensional potential which is periodic in a finite part of the space, and the one which is periodic in entire space. In the limit of the potential extending from −1to1, the results for the potentials periodic in the finite part of space reproduce those for the potentials periodic in entire space. We will see that completely periodic potentials can also be treated invoking symmetry considerations.

Potential Periodic in a Finite Region of Space

Consider a potential V(x) which is periodic with period a in the finite region of space 0 ≤ x ≤ Na such that

whereas on the remaining part of the x-axis,

The solution of the Schrödinger equation corresponding to the potential defined in (12.1) and (12.2) is obtained by solving it in different regions and matching the solutions at the boundaries of the adjoining regions.

We find first the solution in the region 0 ≤ x ≤ Na in which the potential is periodic. Consider the interval na ≤ x ≤ (n + 1)a which we call the nth interval. Since, due to (12.1), the potential in nth interval is same as that in (0, a), the linearly independent solutions of the Schrödinger equation in nth interval are same as those in (0, a).

History is the most fundamental science, for there is no human knowledge which cannot lose its scientific character when men forget the conditions under which it originated, the questions it answered, and the functions it was created to serve.Astudent of quantum mechanics can ignore these words of Benjamin Farrington, often quoted by Schrödinger [1], only at his own peril. For, without their historical perspective, the counter-intuitive nature of the physical laws of the quantum theory may appear to be mystic. Indeed, the laws of quantum mechanics exhibit their distinctive character on a scale which is out of reach of our everyday experience. This is in contrast with the intuitive appeal of the Newton's laws and their conformity with our common experience. However, the inability of the then known theories, referred to now as the classical theories, to explain certain physical phenomena led to the search for a new theory. It culminated with the advent of the quantum theory. The true scientific character of the quantum theory cannot be appreciated without understanding the conditions under which it originated, the questions it answered, and the functions it was created to serve. Such an understanding can even motivate one to ponder whether there can be any other description of the laws of nature which is free from the conceptual and philosophical inconsistencies which are often pointed out in the present-day quantum theory. This chapter outlines brie_y how the ideas leading to the birth of quantum mechanics took shape.