Adventures in Computer Science pp 5374  Cite as
The Principles of Quantum Mechanics
Abstract
Since quantum computing is based on concepts and techniques developed in quantum mechanics, some understanding of the latter is necessary, starting with a history of its development. At the microscopic level, we discuss Louis de Broglie’s proposal that waves behave like particles and particles behave like waves. We then discuss Heisenberg’s uncertainty principle—using a geometrical approach—and explore the Schrödinger equation that underpins the entire framework of quantum mechanics, obtaining several results. Schrödinger’s equation is difficult to interpret, however. A reasonable interpretation of quantum mechanics in probabilistic terms was provided by Max Born. We progress to formulation in quantum mechanics, basing our approach on the axiomatic point of view proposed by Heisenberg and Dirac. To provide a framework for quantum mechanics we list seven postulates, which we define, in turn, in terms of operators. quantum mechanics works with complex numbers, vectors and other entities that are defined in what is called the Hilbert space. We do not explicitly define this space, but introduce the bracket notation of Paul Dirac and describe certain properties—such as the inner product—as groundwork for dealing with concepts in the chapters ahead.
In the 1960s, Rolf Landauer tried to find out whether physical laws imposed limitations on computation. More precisely, he was interested in the reason for energy loss in computers: was it inherent to the laws of physics or due to a defect in the available technology?
One of the problems of current highspeed computers is eliminating the heat produced during computation. On the other hand, as technology evolves, it also increases the scale of integration of electronic devices; thus, for example, more transistors can fit in a given space. Engineers design progressively smaller components with which to build computers. Indeed, we might say that we are already living in the microscopic world. In this context, the restrictions usually imposed by quantum mechanics are extremely important.
It is clear that we cannot build infinitely small computers, since we always need something on which to write and store growing amounts of information. We also need some physical property that allows the writing and storage of such information. In this context, a physical property of some microscopic systems (such as the atom), called spin , can be used for our purposes. Systems with natural spin have measurable physical attributes to which we can assign numbers in such a way that each number represents a state. But before we go any further, we need to make a historical detour into the field of Quantum Mechanics.
4.1 History and Basic Concepts of Quantum Mechanics
Going back in time to the nineteenth century, certain experiments and the discovery of natural radioactivity confirmed that atoms and molecules are, at least in part, composed of charged particles. In 1909, Ernest Rutherford demonstrated the nuclear nature of atoms by observing the deflection of alpha particles passing through a thin gold foil. The results of this experiment underpinned the hypothesis of the planetary structure of atoms, according to which atoms consist of a dense nucleus formed of neutrons and protons, with electrons orbiting very far from the nucleus. The properties of atoms, motion and energy thus depend on their electronic structure.
However, the planetary model of the atom is neither compatible nor consistent with James Clerk Maxwell ’s classical electromagnetic theory, which predicts that charged particles, when accelerated, will lose energy. Obviously the electron is an accelerated particle, because the direction of its velocity changes constantly. This circumstance inevitably leads to a loss of energy by radiation. In other words, according to the planetary model, atoms would not be stable, as the electron would draw a spiral orbit around the nucleus that would finally lead to its collapse. But this is not what happens, as electrons remain stable.
Going into more detail, for the moment assume that we have an electron e^{−} with mass m and moving at velocity v, and also assume that λ is the associated wavelength, defined by

h is the Planck constant

λ(e) is the electron wavelength

m(e) is the electron mass

p(e) is the linear momentum of the electron

v(e) is the electron velocity
And, in accordance with Planck, the total energy of a wave is therefore
In the previous equation, ν is the frequency of the wave. Therefore
But since v(e) < c
Some photon and electron physical properties
Physical entity  Type  Velocity  Mass  Treatment 

Photon  Particle of light  c in vacuum  None  Always relativistic 
Electron  Wave of matter  Less than c  >0  Not always relativistic 
4.2 Heisenberg’s Uncertainty Principle

How can we describe the position of a particle?

What is the right procedure for determining this position?
The obvious answer to both questions would seem to be “by observing the particle.” But in the world of very small things, observing is not that easy. Imagine a microscope that can be used to observe an electron. To see the electron , we have to project a beam of light or some kind of appropriate radiation onto it. But an electron is so small that even a single photon of light would change its position. So in the precise instant we want to measure the position, the position changes.
Heisenberg demonstrated that it was not possible to contrive a method to locate the position of a subatomic particle unless we admitted some absolute uncertainty with regard to its exact velocity, since it is impossible to simultaneously and accurately measure both position and velocity. This is also the reason why we cannot achieve complete absence of energy, even at the absolute zero point. Particles at the zero point with absolutely no energy would remain totally motionless (zero velocity), so their position could easily be measured. However, in accordance with Heisenberg’s uncertainty principle, we have to expect some zeropoint energy.
In 1930, Einstein showed that the uncertainty principle—which affirms the impossibility of reducing error in the measurement of position without increasing error in the measurement of linear momentum—also implied the impossibility of reducing error in the measurement of energy without increasing uncertainty in the measurement of time. For a particle moving in direction x, if we express uncertainty in both position and linear momentum as
and if we express uncertainty in both energy and time as
then a very simple dimensional analysis shows that both uncertainties have the same dimensional result: M × L^{2} × T^{−1}, where M is mass, L is length and T is time.
 1.
A box is hanging from a spring.
 2.
A clock is hanging from the box.
 3.
The box has a small hole.
 4.
A particle is moving inside the box.
 5.
A ruler measures elongation of the spring.
 6.
An observer is looking simultaneously at the spring, clock and ruler.
 7.
The particle exits the box through the hole.
 8.
The observer knows exactly when this happens because the spring shortens at a time indicated by the clock.
Einstein’s satisfaction at disproving the Heisenberg uncertainty principle only lasted until Niels Bohr demonstrated, using Einstein’s own theory of general relativity, that the principle was correct. When the particle exits the box, the spring shortens, and this is how the observer knows that the particle is outside the box. However, the gravitational field acting on the clock is now different, so time runs differently compared to the initial state.
As a consequence of the wavelike properties of matter, the beam of particles diffracts, causing the momentum to be changed by a given angle α (in other words, part of the momentum is transferred elsewhere). Formally
There is a slit in the axis OX.
The slit diameter is W.
p is the momentum vector.
p is the modulus of p.
Δx = W is the uncertainty of the position.
Δp _{ x } = p × sin(α) is the uncertainty in the momentum.
Δx × Δp _{ x } = W × p × sin(α)
 1.
We calculate the value of α corresponding to the first diffraction minimum.
 2.
We draw AC such that AD = CD, with OC representing the difference between the distances traveled by the particles going through A and through O.
 3.
The distance OD is large relative to W, so AD and OD are almost parallel, meaning that the angle ACO is almost a right angle. Hence, OAC = α.
 4.
Steps 2 and 3 imply that
 5.
If we say that OC = λ/2, then we have that
Recall now that
However, since the uncertainties have not been defined with precision, we have to conclude that
From the above result we can conclude that the simple fact of measuring causes a change in the state of a system which is of the order of magnitude of the Planck constant h.
4.3 Schrödinger’s Equation
In 1925 Erwin Schrödinger developed an equation that describes how a nonrelativistic massive particle changes with time. Schrödinger’s equation—of central importance in quantum mechanics—represents for microscopic particles what Newton’s second law of motion represents for classical mechanics. For a onedimensional system of a single particle, Newton’s second law establishes that
Then
Assuming that
if the particle is in position x_{0} at an arbitrary initial time t_{0,} then
Recall that the velocity v of a moving particle can be represented as
If the particle is moving with velocity v_{0} at an arbitrary initial time t_{0}, then

F is the net force

m is the particle mass

x is the particle position

t is time

g is a generic function

v is the particle velocity

c_{1} and c_{2} are integration constants
We can now easily calculate c_{1} and c_{2}. In classical mechanics, if we know x, v and F we can predict the state of a particle in motion. However, matters are somewhat different in the microscopic world. In quantum mechanics the state of a system is described by means of a wave function:
Bearing in mind the wavelike behavior of small particles, the starting point for developing an equation to describe waves of matter is the equation developed by Louis de Broglie :
As a first restriction, the equation we want to define must be consistent with de Broglie’s equation. Our second restriction, given the wavelike properties of microscopic particles, will be that the mathematical formula to describe the behavior of a particle with wavelength λ must be a trigonometric function. For example,
We also have to take into account that the total energy E of the particle is the sum of its kinetic energy K and its potential energy V:
However
Thus
And
To obtain a differential equation with a sinusoidal solution (the case of the classical approach), we have to differentiate and make some substitutions. Thus
This last equation is valid when potential energy V is constant. However, let us consider the above equation to be also valid when V is not constant. Conversely, V can vary along the xaxis. In mathematical terms this is represented as
We can thus obtain Schrödinger’s equation independent of time:
If we include time in the potential energy of our system, then the basic equation reflecting a onedimensional quantum system for a single timedependent particle is as follows:
It thus follows that, given
then
In this equation , V(x, t) is the potential energy of our quantum system. But Schrödinger’s equation is not only difficult to interpret, but also (with the exception of some very rare quantum systems) impossible to solve analytically. There is also another question: what is the real physical sense of the Schrödinger equation? Heisenberg indicated that the classical state and the quantum state have to be interpreted differently since they are concerned with different things. The problem of the physical interpretation of Schrödinger’s equation was finally resolved by the mathematician Max Born . For a given time t,
 a.
N is a collection of identical systems that do not interact with each other
 b.
∀i ∈ N, ∀j ∈ N: Ψi = Ψj
 c.
The position of each i ∈ N is measured
 d.
dn_{ x } = number of particles in the interval (x, x + dx)
 e.
dn_{ x }/N = probability = P(x, x + dx) = Ψ^{2} dx
However , we cannot make many measurements of a single particle since, in accordance with Heisenberg’s uncertainty principle, each measurement changes the state of the system.
4.4 Schrödinger’s Equation Revisited
Let us approach the same problem from a different point of view and obtain a similar result. Our starting point will be the timedependent Schrödinger equation:
Then
And
Separating variables and operating
Dividing this last expression by ψ(x)f(t) we obtain
Now we have a term that is independent of x:
Since the two terms are independent and are related by equality, they must represent a constant magnitude (energy, perhaps?). So, if we denote this constant magnitude by E, the result is
We can now operate with the term that is independent of x as follows:
And
Therefore
In this last equation, the arbitrary constant A = e^{cte} can be omitted since it does not affect the generality of the formula. We therefore conclude that
Analogously, if we operate with the term that is independent of time t
Tidying up we obtain the following equation:
We have seen this equation before. It is, in fact, the onedimensional, timeindependent Schrödinger equation, where E represents the total energy in the system. Recalling the equation that defines f(t), we can obtain the wave function for the probability density steady state as follows:
Note that quantum equations work with complex numbers, which are also at the heart of quantum computing. This issue will be discussed later in some detail.
4.5 The Postulates of Quantum Mechanics
To understand quantum computing we need to understand some elementary principles of quantum mechanics. It was not possible to properly describe the behavior of very small particles such as electrons, atoms or molecules until physicists (among them Niels Bohr , Max Born , Louis de Broglie , Paul Dirac , Albert Einstein , Werner Heisenberg , Pascual Jordan , Wolfgang Pauli and Erwin Schrödinger ) mathematically formalized the theory of quantum mechanics in the closing years of the 1920s.

The strange behavior of blackbody spectral radiation

The photoelectric effect

The atomic spectrum of the hydrogen atom

Compton scattering
The second approach is axiomatic: we define fundamental postulates from which we deduce the behavior of microscopic physical systems and then compare the theory with experimental observations . By quantifying the level of agreement between theoretical predictions and experimental results we can obtain a direct measure of the validity of the theory.
We will use the second, axiomatic, approach to quantum mechanics. The bestknown phenomenological formulation is that by Schrödinger, based on the wavelike description of matter. However, we will use the formulation of Heisenberg and Dirac, based on vectors, operators and matrices. Genius is genius: Schrödinger demonstrated that the two approaches are equivalent and interchangeable.
The axiomatic formulation of Heisenberg and Dirac is based on the postulates described below.
Postulate I
The state of a physical system is described by a function Ψ(q, t) of coordinates (q) and time (t). This function, called the state function or wave function, contains all the information that can be obtained on the system in question.
Postulate II
The temporal evolution of the state of the system is given by the timedependent Schrödinger equation:
In this equation, ℏ = h/2π and \( \widehat{\mathbf{H}} \) is the Hamiltonian operator. For a single particle moving along axis x, \( \widehat{\mathbf{H}} \) is given by
Postulate III
For each observable (physical) property, quantum mechanics defines a linear Hermitian operator. To find this operator, we have to write the classical expression of the observable property in terms of the Cartesian coordinates and the corresponding linear momentum. We then have to replace each x coordinate by the operator x (that is, multiply by x) and replace each linear momentum p_{ x } by the operator −iℏd/dx.
Postulate IV
Independently of the state function of the system, the only values that we can obtain from a measurement of the observable A are the eigenvalues a of
Postulate V
If \( \widehat{A} \) is a linear Hermitian operator that represents a given observable, then the eigenfunctions ψ_{ i } of
form a complete set. This last statement means that any state function Ψ that satisfies the selfsame conditions can be represented as the linear combination of the eigenstates of A:
Postulate VI
If ψ_{ i }(q, t) is the normalized state function of the system at a given time t, then the mean value of the observable A at time t is:
where the symbol (*) means conjugate (remember that we are working with complex numbers).
Postulate VII
If \( \widehat{A} \) is a linear Hermitian operator that represents a given physical magnitude, then the eigenfunctions f_{ i } of the operator \( \widehat{A} \) form a complete set.
4.6 Some Quantum Operators
An operator is a rule or procedure that, given one function, allows us to calculate another function. What follows is a list of operations and their properties applying to quantum operators, denoted here by means of capital letters with a hat (that is, \( \widehat{A} \), \( \widehat{E} \), \( \widehat{I} \) etc.)
We need to take into account that, generally speaking
We can also define an algebra of quantum operators using the following elements: Assume we have two operators \( \widehat{A} \) and \( \widehat{E} \)

The identity operator is defined.

The null operator is defined. It holds that

Not always \( \widehat{A}\times \widehat{E}=\widehat{E}\times \widehat{A} \)

The commutator of two operators is defined:

\( \widehat{A}\left[f(x)+g(x)\right]=\widehat{A}f(x)+\widehat{A}g(x) \), where f and g are functions

If f is a function and c is a constant: \( \widehat{A}\left[c\ f(x)\right]=c\ \widehat{A}\ f(x) \)
In the quantum approach all operators are linear.
Other important characteristics of quantum operators are related to the concepts of eigenfunctions and eigenvalues (to be discussed further below). Assume that \( \widehat{A} \) is an operator, f(x) is a function and k is a constant. If
then f(x) is an eigenfunction and k is an eigenvalue of the operator \( \widehat{A} \). The eigenfunctions of any linear operator also verify that, given a constant c,
Thus:
If f(x) is an eigenfunction of \( \widehat{A} \) with eigenvalue k, then c × f(x) is also an eigenfunction.
Also associated with the quantum operators is the notion of average value. Consider a physical magnitude E. When the state function Ψ is not an eigenfunction of the associated operator \( \widehat{E} \), measuring E gives us just one possible value. We now consider the average value of E for a system whose state is Ψ. Since the experimental result of the average value of E forces us to consider a large number of systems, all in the same state Ψ, we need to measure E for each system. The average value of E is the arithmetical average of the observed values. For example, if e_{1}, e_{2,}… are the values observed for E, then the average value of E, denoted by 〈E〉, for a large number N of systems is as follows:
The same result can be obtained if we sum all the possible values of E, that is to say, if we sum the different e_{ i } that can be obtained, multiplying each by the number of times that it has been observed:
In this last expression, P_{ e } is the probability of observing the value e, since, as we have said, N is very large. We can use these results to study the behavior of a onedimensional particle system in, for example, the state Ψ(x, t). More specifically, we consider the average value for the coordinate x.
Let us assume a given particle, which could perfectly well be a bit, in the state Ψ(x, t). Let us also assume that this particle is moving along the xaxis in such a way that x takes continuous (nondiscrete) values. Under these assumptions,
is the probability of observing the particle under consideration between (x) and (x + dx). Therefore
On the other hand, if E(x) is a property that depends on x, and if \( \widehat{E} \) is the operator associated with E(x), then
The associated probability density is defined as follows:

〈F + G〉 = 〈F〉 + 〈G〉

〈F × G〉 ≠ 〈F〉 × 〈G〉
A problem with this approach is that the mathematical background involves numerous integrals that can make understanding the approach difficult. The solution to this notation problem was provided by Paul Dirac .
Let us assume that τ indicates the integration space. If φ_{ m } and φ_{ n } are functions and \( \widehat{A} \) is an operator, then
In the above equations, 〈m  n〉 is the Dirac representation of the inner product between the vectors m and n. Using Dirac braket notation, for a given state described as a row vector or a column vector, then the bra notation or the ket notation, respectively, can be used:
The inner product for m and n is as follows.
Then
Let us illustrate this with an example. Assume that
Then

\( {\left\langle m\mid n\right\rangle}^{\ast }=\left\langle n\mid m\right\rangle \)

\( {\left\langle m\mid m\right\rangle}^{\ast }=\left\langle m\mid m\right\rangle \)
These are verified as follows:
And
As mentioned earlier, it is not our purpose to comprehensively describe quantum mechanical principles. However, the concepts described above are probably sufficient to understand the basic principles of quantum computing.
4.7 Chapter Summary
Since quantum computing is based on concepts and techniques developed in quantum mechanics, some understanding of the latter is necessary, starting with a history of its development. At the microscopic level, we discuss Louis de Broglie’s proposal that waves behave like particles and particles behave like waves. We then discuss Heisenberg’s uncertainty principle—using a geometrical approach—and explore the Schrödinger equation that underpins the entire framework of quantum mechanics, obtaining several results. Schrödinger’s equation is difficult to interpret, however. A reasonable interpretation of quantum mechanics in probabilistic terms was provided by Max Born. We progress to formulation in quantum mechanics, basing our approach on the axiomatic point of view proposed by Heisenberg and Dirac. To provide a framework for quantum mechanics we list seven postulates, which we define, in turn, in terms of operators. Quantum mechanics works with complex numbers, vectors and other entities that are defined in what is called the Hilbert space. We do not explicitly define this space, but introduce the bracket notation of Paul Dirac and describe certain properties—such as the inner product—as groundwork for dealing with concepts in the chapters ahead.
4.8 Glossary of Terms and Notation Used in This Chapter
Terms

Absolute zero point: the point at which the fundamental particles of nature have minimal vibrational motion, retaining only quantum mechanical, zeropointenergyinduced particle motion.

Alpha particle: positively charged particle consisting of two protons and two neutrons, emitted in radioactive decay or nuclear fission. Also the nucleus of a helium atom.

Average value: a quantity intermediate to a set of quantities.

Black body: a hypothetical body that absorbs but does not reflect electromagnetic radiation incident on its surface.

Complex number: a mathematical expression (a + bi) in which a and b are real numbers and i^{2} = −1.

Differential equation: an equation involving differentials or derivatives.

Diffraction: the spreading of waves around obstacles.

Eigenfunction: any nonzero function f for a linear operator A defined in some function space that returns from the operator exactly as is, except for a multiplicative scaling factor.

Eigenstate: a quantummechanical state corresponding to an eigenvalue of a wave equation.

Eigenvalue: a special set of scalars associated with a linear system of equations (that is, a matrix equation) that are sometimes also known as characteristic roots, characteristic values, proper values or latent roots.

General relativity: a generalization of special relativity and Newton’s law of universal gravitation that provides a unified description of gravity as a geometric property of space and time (or spacetime). The curvature of spacetime is directly related to the energy and momentum of the matter and radiation that are present.

Hamiltonian: the operator (in most cases) corresponding to the total energy of the system.

Heisenberg’s uncertainty principle: in Quantum Mechanics, any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle (complementary variables), such as position x and momentum p, can be known simultaneously.

Hermitian: in mathematics, a selfadjoint matrix or square matrix with complex entries that is equal to its own conjugate transpose. Thus, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j. In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign.

Hilbert space: an abstract vector space possessing the structure of an inner product that allows length and angle to be measured.

Inner product space: in linear algebra, a vector space with an additional structure called an inner product that associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as vector length and the angle between two vectors and also provide the means of defining orthogonality between vectors (zero inner product).

Linear momentum: the product of the mass and velocity of an object.

Matrix: in mathematics, a rectangular array of numbers, symbols or expressions, arranged in rows and columns.

Operator: a mapping from one vector space or module to another. Operators are of critical importance in both linear algebra and functional analysis, and have applications in many other fields of pure and applied mathematics. For example, in classical mechanics, the derivative is used ubiquitously, and in quantum mechanics, observables are represented by Hermitian operators. Important properties of operators include linearity, continuity and boundedness.

Photoelectric effect: the observation that light shone on many metals produces electrons called photoelectrons. The effect is commonly studied in electronic physics and in chemical disciplines such as quantum chemistry and electrochemistry.

Photon: a particle representing a quantum of light or other electromagnetic radiation.

Planck’s constant: a physical constant that is the quantum of action and central to quantum mechanics.

Probability density: a function that describes the relative likelihood of a random variable taking on a given value.

Spin: in quantum mechanics and particle physics, an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons) and atomic nuclei.

Trigonometric (circular) function: in mathematics, any function of an angle that relates the angles of a triangle to the lengths of its sides.

Wave function: a quantum state of an isolated system of one or more particles, whereby one wave function contains all information on the entire system (not separate wave functions for each particle in the system). Its interpretation is that of a probability amplitude and it can be used to derive quantities associated with measurements, such as the average momentum of a particle.
Notation
 λ

Wavelength
 h

Planck constant
 c

Speed of light in a vacuum
 ν

Frequency
 Ψ(x, t)

Wave function
 \( \widehat{H} \)

Hamilton operator or Hamiltonian
 \( \widehat{A} \)

A generic operator
 \( \left[\widehat{A},\widehat{E}\right] \)

Commutator of two operators
 〈〉

Average value
 m

A vector m