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The Principles of Quantum Mechanics

  • Vicente Moret-Bonillo
Chapter

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 high-speed 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.

In the first decade of the twentieth century, Max Planck was trying to solve the apparently strange behavior of energy in what came to be called the black-body radiation problem. Albert Einstein , meanwhile, was studying the photoelectric effect, which reveals the corpuscular character of light. Both their contributions made the quantum nature of the universe evident. According to this new conception of the universe, the stability of atoms can be explained if we assume stationary states in the electron orbits, as depicted in Fig. 4.1.
Fig. 4.1

Quantum nature of the universe. Emission and absorption of discrete quantities of energy between two allowed energy states

In 1923 Louis de Broglie demonstrated that electrons also have wave-like behavior. Surprisingly, therefore, particles can behave like waves and waves can behave like particles. Figure 4.2 illustrates these behaviors with an example that involves photons (particles of light), where is energy, h is Planck’s constant, ν is frequency, and electrons are represented as e. Figure 4.3 is a photographic image that reflects this duality.
Fig. 4.2

The dual particle-wave nature of light: left, photons behaving as particles, and right, photons behaving as waves

Fig. 4.3

An image showing particle-wave duality. In the microscopic world, all depends on how we look at something, but a cylinder remains a cylinder

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

$$ \lambda \left(\mathrm{e}\right)=\frac{h}{m\left(\mathrm{e}\right)\times v\left(\mathrm{e}\right)}=\frac{h}{p\left(\mathrm{e}\right)} $$
In this equation
  • 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

Assume now that v(e) = c, where c is the speed of light in a vacuum . In accordance with Einstein, the total energy of a particle (in our case, an electron e) is therefore
$$ E=m\times {c}^2 $$

And, in accordance with Planck, the total energy of a wave is therefore

$$ E=h\times v $$

In the previous equation, ν is the frequency of the wave. Therefore

$$ m\times {c}^2=h\times \nu =\frac{h\times c}{\lambda}\to \lambda =\frac{h}{m\times c} $$

But since v(e) < c

$$ \lambda \left(\mathrm{e}\right)=\frac{h}{m\left(\mathrm{e}\right)\times v\left(\mathrm{e}\right)}=\frac{h}{p\left(\mathrm{e}\right)} $$
In view of these results , in 1932 Otto Stern postulated that the wave-like effects were the result of a general law of motion. Almost at the same time that these ideas were being consolidated, engineers started to consider new applications. Electronic microscopy, for instance, is a practical application of the dual nature of particles. However, despite this dual nature, photons (particles of light) are not the same as electrons (waves of matter). The difference is summarized in Table 4.1.
Table 4.1

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

We will now look at one of the most important, and controversial, concepts in Quantum Mechanics: the uncertainty principle introduced by Werner Heisenberg . The questions we need to answer are
  • 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 zero-point 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

$$ \Delta {p}_x\times \Delta x $$

and if we express uncertainty in both energy and time as

$$ \Delta E\times \Delta t $$

then a very simple dimensional analysis shows that both uncertainties have the same dimensional result: M × L2 × T−1, where M is mass, L is length and T is time.

In an attempt to demonstrate that the Heisenberg uncertainty principle was wrong, Einstein proposed the experiment depicted in Fig. 4.4.
Fig. 4.4

Einstein’s experiment to intend to demonstrate that the Heisenberg uncertainty principle was wrong

Einstein’s experiment is described as follows:
  1. 1.

    A box is hanging from a spring.

     
  2. 2.

    A clock is hanging from the box.

     
  3. 3.

    The box has a small hole.

     
  4. 4.

    A particle is moving inside the box.

     
  5. 5.

    A ruler measures elongation of the spring.

     
  6. 6.

    An observer is looking simultaneously at the spring, clock and ruler.

     
  7. 7.

    The particle exits the box through the hole.

     
  8. 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.

Returning to the fundamentals of the uncertainty principle, although formal and rigorous demonstrations exist, we will prove it using a geometrical approach. Consider the diagram in Fig. 4.5, which shows a beam of particles hitting a wall with a small slit whose diameter is W. Situated behind the slit is photographic film, which will detect motion.
Fig. 4.5

An experiment to illustrate Heisenberg’s uncertainty principle

As a consequence of the wave-like 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)

In accordance with the condition of the first diffraction minimum , the difference between the distance traveled by the particles going through A and going through O has to be one half of the associated wavelength (λ/2), as illustrated in Fig. 4.6.
Fig. 4.6

A geometrical outline of Heisenberg’s uncertainty principle

The proof is as follows:
  1. 1.

    We calculate the value of α corresponding to the first diffraction minimum.

     
  2. 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. 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. 4.

    Steps 2 and 3 imply that

     
$$ \mathrm{OC}=\frac{W}{2}\times \sin \left(\alpha \right) $$
  1. 5.

    If we say that OC = λ/2, then we have that

     
$$ \lambda =W\times \sin \left(\alpha \right) $$
We previously stated that
$$ \Delta x\times \Delta {p}_x=W\times p\times \sin \left(\alpha \right)\to \Delta x\times \Delta {p}_x=p\times \lambda $$

Recall now that

$$ \lambda =\frac{h}{p}\to \Delta x\times \Delta {p}_x=h $$

However, since the uncertainties have not been defined with precision, we have to conclude that

$$ \Delta x\times \Delta {p}_x\approx h $$

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 non-relativistic 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 one-dimensional system of a single particle, Newton’s second law establishes that

$$ F=m\frac{{\mathrm{d}}^2x}{\mathrm{d}{t}^2} $$

Then

$$ {\mathrm{d}}^2x=F\times {m}^{-1}\times \mathrm{d}{t}^2 $$

Assuming that

$$ x=g\left(t,{c}_1,{c}_2\right) $$

if the particle is in position x0 at an arbitrary initial time t0, then

$$ {x}_0=g\left({t}_0,{c}_1,{c}_2\right) $$

Recall that the velocity v of a moving particle can be represented as

$$ v=\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\left\{\ g\left(t,{c}_1,{c}_2\right)\ \right\} $$

If the particle is moving with velocity v0 at an arbitrary initial time t0, then

$$ {v}_0=\frac{\mathrm{d}}{\mathrm{d}t}{\left\{\ g\left(t,{c}_1,{c}_2\right)\ \right\}}_{t={t}_0}\kern0.5em $$
In the above equations
  • 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

  • c1 and c2 are integration constants

We can now easily calculate c1 and c2. 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:

$$ \varPsi =\varPsi \left(x,t\right) $$

Bearing in mind the wave-like behavior of small particles, the starting point for developing an equation to describe waves of matter is the equation developed by Louis de Broglie :

$$ p=m\times v=\frac{h}{\lambda } $$

As a first restriction, the equation we want to define must be consistent with de Broglie’s equation. Our second restriction, given the wave-like 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,

$$ \Psi (x)=\sin \left(\frac{2\pi x}{\lambda}\right) $$

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:

$$ E=K+V $$

However

$$ K=\frac{1}{2}m\times {v}^2 $$

Thus

$$ \frac{1}{2}m\times {v}^2+V=E $$

And

$$ {\displaystyle \begin{array}{c}v=\frac{h}{m\times \lambda}\to \frac{h^2}{2\times m\times {\lambda}^2}=E-V\\ {}\frac{1}{\lambda^2}=\frac{2\times m}{h^2}\left(E-V\right)\end{array}} $$

To obtain a differential equation with a sinusoidal solution (the case of the classical approach), we have to differentiate and make some substitutions. Thus

$$ \frac{\mathrm{d}}{\mathrm{d}x}\Psi (x)=\frac{2\pi }{\lambda}\cos \left(\frac{2\pi x}{\lambda}\right) $$
$$ \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\Psi (x)=-{\left(\frac{2\pi }{\lambda}\right)}^2\sin \left(\frac{2\pi x}{\lambda}\right) $$
$$ \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\Psi (x)=-{\left(\frac{2\pi }{\lambda}\right)}^2\Psi (x) $$
$$ \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\Psi (x)=-\frac{8{\pi}^2m}{h^2}\left(E-V\right)\ \Psi (x) $$

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 x-axis. In mathematical terms this is represented as

$$ V=V(x) $$

We can thus obtain Schrödinger’s equation independent of time:

$$ \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\Psi (x)=-\frac{8{\pi}^2m}{h^2}\ \left[E-V(x)\right]\ \Psi (x) $$

If we include time in the potential energy of our system, then the basic equation reflecting a one-dimensional quantum system for a single time-dependent particle is as follows:

$$ \frac{-\hslash }{i}\frac{\mathrm{d}\Psi \left(x,t\right)}{\mathrm{d}t}=\frac{-{\hslash}^2}{2m}\frac{{\mathrm{d}}^2\Psi \left(x,t\right)}{\mathrm{d}{x}^2}+V\left(x,t\right)\Psi \left(x,t\right) $$

It thus follows that, given

$$ \hslash =\frac{h}{2\pi } $$

then

$$ \frac{-h}{2\pi i}\frac{\mathrm{d}\Psi \left(x,t\right)}{\mathrm{d}t}=\frac{-{h}^2}{8{\pi}^2m}\frac{{\mathrm{d}}^2\Psi \left(x,t\right)}{\mathrm{d}{x}^2}+V\left(x,t\right)\Psi \left(x,t\right) $$

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,

$$ {\left|\Psi \left(x,t\right)\right|}^2 $$
is the probability density function. Probability density, in Quantum Mechanics, refers to the probability that a particle is in the interval (x, x + dx). If g(x) is the probability density, with g(x) ∈ R such that g(x) ≥ 0, then, in accordance with Fig. 4.7,
Fig. 4.7

An attempt to introduce the concept of probability density

$$ g(x)={\left|\Psi \right|}^2\to \mathrm{Prob}(x)={\int}_a^b{\left|\Psi \right|}^2\mathrm{d}x $$
$$ {\int}_{-\infty}^{\infty }{\left|\Psi \right|}^2\mathrm{d}x=1 $$
We can now establish an interesting relationship between |Ψ|2 and experimental measurements:
  1. a.

    N is a collection of identical systems that do not interact with each other

     
  2. b.

    i ∈ N, ∀j ∈ N: Ψi = Ψj

     
  3. c.

    The position of each i ∈ N is measured

     
  4. d.

    dn x  = number of particles in the interval (x, x + dx)

     
  5. 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 time-dependent Schrödinger equation:

$$ \frac{-h}{2\times \pi \times i}\ \frac{\mathrm{d}\ \Psi \left(x,t\right)}{\mathrm{d}t}=\frac{-{h}^2}{8\times {\pi}^2\times m}\ \frac{{\mathrm{d}}^2\ \Psi \left(x,t\right)}{\mathrm{d}{x}^2}+V\left(x,t\right)\Psi \left(x,t\right) $$
In the words of Ira Levine , this equation is no less than impressive . Nevertheless, there are situations in which potential energy does not vary over time. In such cases we can experiment to see what happens. Assume that
$$ \Psi \left(x,t\right)=f(t)\times \uppsi (x) $$

Then

$$ \frac{\mathrm{d}\uppsi \left(x,t\right)}{\mathrm{d}t}=\psi (x)\frac{\mathrm{d}f(t)}{\mathrm{d}t} $$

And

$$ \frac{{\mathrm{d}}^2\Psi \left(x,t\right)}{\mathrm{d}{x}^2}=f(t)\frac{{\mathrm{d}}^2\psi (x)}{\mathrm{d}{x}^2} $$

Separating variables and operating

$$ \frac{-h}{2\times \pi \times i}\psi (x)\frac{\mathrm{d}f(t)}{\mathrm{d}t}=\frac{-{h}^2}{8\times {\pi}^2\times m}\ f(t)\ \frac{{\mathrm{d}}^2\psi (x)}{\mathrm{d}{x}^2}+V(x)\psi (x)f(t) $$

Dividing this last expression by ψ(x)f(t) we obtain

$$ \frac{-h}{2\times \pi \times i}\ {f}^{-1}(t)\frac{\mathrm{d}f(t)}{\mathrm{d}t}=\frac{-{h}^2}{8\times {\pi}^2\times m}{\psi}^{-1}(x)\frac{{\mathrm{d}}^2\psi (x)}{\mathrm{d}{x}^2}+V(x) $$

Now we have a term that is independent of x:

$$ \frac{-h}{2\times \pi \times i}\ {f}^{-1}(t)\frac{\mathrm{d}f(t)}{\mathrm{d}t} $$
And we also have a term that is independent of t:
$$ \frac{-{h}^2}{8\times {\pi}^2\times m}{\psi}^{-1}(x)\frac{{\mathrm{d}}^2\psi (x)}{\mathrm{d}{x}^2}+V(x) $$

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

$$ E=\frac{-h}{2\times \pi \times i}\ {f}^{-1}(t)\frac{\mathrm{d}f(t)}{\mathrm{d}t}=\frac{-{h}^2}{8\times {\pi}^2\times m}{\psi}^{-1}(x)\frac{{\mathrm{d}}^2\psi (x)}{\mathrm{d}{x}^2}+V(x) $$

We can now operate with the term that is independent of x as follows:

$$ {f}^{-1}(t)\mathrm{d}f(t)=-\frac{2\times \pi \times i}{h}E\times \mathrm{d}t $$

And

$$ {\log}_{\mathrm{e}}f(t)=-\frac{2\times \pi \times i}{h}E\times t+\mathrm{cte} $$

Therefore

$$ f(t)={\mathrm{e}}^{\mathrm{cte}}\times {\mathrm{e}}^{-\frac{2\times \pi \times i}{h}E\times t}=A\times {\mathrm{e}}^{-\frac{2\times \pi \times i}{h}E\times t} $$

In this last equation, the arbitrary constant A = ecte can be omitted since it does not affect the generality of the formula. We therefore conclude that

$$ f(t)={\mathrm{e}}^{-\frac{2\times \pi \times i}{h}E\times t} $$

Analogously, if we operate with the term that is independent of time t

$$ E\psi (x)=\frac{-{h}^2}{8\times {\pi}^2\times m}\times \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\psi (x)+V(x)\psi (x) $$

Tidying up we obtain the following equation:

$$ \frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}\psi (x)+\frac{8{\pi}^2m}{h^2}\left[E-V(x)\right]\psi (x)=0 $$

We have seen this equation before. It is, in fact, the one-dimensional, time-independent 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:

$$ f(t)={\mathrm{e}}^{-\frac{2\times \pi \times i}{h}E\times t} $$
$$ \mathrm{If}:V=V(x)\ \overset{\mathrm{yields}}{\to }\ \varPsi \left(x,t\right)=\psi (x)\times {e}^{-\frac{2\times \pi \times i}{h}E\times t} $$

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.

Quantum Mechanics can be approached in two different ways. The first way is to analyze physical problems that quantum mechanics but not classical mechanics can resolve, for instance:
  • The strange behavior of black-body 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 best-known phenomenological formulation is that by Schrödinger, based on the wave-like 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 time-dependent Schrödinger equation:

$$ i\mathit{\hslash}\frac{d\Psi \left(q,t\right)}{\mathrm{d}t}=\widehat{H}\ \Psi \left(q,t\right) $$

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

$$ \widehat{H}=-\frac{\hslash^2}{2m}\ \frac{d\Psi \left(q,t\right)}{\mathrm{d}x}+V\left(x,t\right)\Psi \left(x,t\right) $$

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

$$ \widehat{A}\ {f}_i=a\times {f}_i $$

Postulate V

If \( \widehat{A} \) is a linear Hermitian operator that represents a given observable, then the eigenfunctions ψ i of

$$ \widehat{A}\ {\psi}_i=a\times {\psi}_i $$

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:

$$ \varPsi =\sum \limits_i{c}_i{\psi}_i $$

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:

$$ \left\langle A\right\rangle =\int {\psi}^{\ast}\widehat{A}\ \psi\ \mathrm{d}q $$

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.)

$$ \left(\widehat{A}+\widehat{E}\right)\ f(x)=\widehat{A}f(x)+\widehat{E}f(x) $$
$$ \left(\widehat{A}\times \widehat{E}\right)\ f(x)=\widehat{A}\left\{\ \widehat{E}f(x)\right\} $$

We need to take into account that, generally speaking

$$ \left(\widehat{A}\times \widehat{E}\right)\ f(x)\ne \left(\widehat{E}\times \widehat{A}\right)\ f(x) $$

We can also define an algebra of quantum operators using the following elements: Assume we have two operators \( \widehat{A} \) and \( \widehat{E} \)

$$ \widehat{A}=\widehat{E}\leftrightarrow \forall f\widehat{A}f=\widehat{E}f $$
  • The identity operator is defined.

  • The null operator is defined. It holds that

$$ \widehat{A}\left(\widehat{E}\times \widehat{I}\right)=\left(\widehat{A}\times \widehat{E}\right)\widehat{I} $$
  • Not always \( \widehat{A}\times \widehat{E}=\widehat{E}\times \widehat{A} \)

  • The commutator of two operators is defined:

$$ \left[\widehat{A},\widehat{E}\right]=\widehat{A}\times \widehat{E}-\widehat{E}\times \widehat{A} $$
$$ {\widehat{A}}^2=\widehat{A}\times \widehat{A} $$
Quantum operators have some interesting properties:
  • \( \widehat{A}\left[f(x)+g(x)\right]=\widehat{A}f(x)+\widehat{A}g(x) \), where f and g are functions

  • If f is a function and c is a constant: \( \widehat{A}\left[c\ f(x)\right]=c\ \widehat{A}\ f(x) \)

$$ \left(\widehat{A}+\widehat{E}\right)\ \widehat{I}=\widehat{A}\widehat{I}+\widehat{E}\widehat{I} $$
$$ \widehat{A}\left(\widehat{E}+\widehat{I}\right)=\widehat{A}\widehat{E}+\widehat{A}\widehat{I} $$

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

$$ \widehat{A}f(x)=k\times f(x) $$

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,

$$ \widehat{A}\left(c\times f\right)=c\times \widehat{A}f=c\times k\times f=k\left(c\times f\right) $$

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 e1, e2,… 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:

$$ \left\langle E\right\rangle =\frac{\sum_{i=1}^N{e}_i}{N} $$

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:

$$ \left\langle E\right\rangle =\frac{\sum_e{n}_e\times e}{N}=\sum \limits_e\left(\frac{n_e}{N}\right)\times e=\sum \limits_e{P}_e\times e $$

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 one-dimensional 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 x-axis in such a way that x takes continuous (non-discrete) values. Under these assumptions,

$$ {P}_{\mathrm{bit}}\left(x,x+\mathrm{d}x\right)={\left|\Psi \right|}^2\mathrm{d}x $$

is the probability of observing the particle under consideration between (x) and (x + dx). Therefore

$$ \left\langle x\right\rangle =\underset{-\infty }{\overset{+\infty }{\int \limits }}x\ {\left|\Psi \left(x,t\right)\right|}^2\ \mathrm{d}x $$

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

$$ \left\langle E\right\rangle =\int {\varPsi}^{\ast }E\ \Psi\ \mathrm{d}x $$

The associated probability density is defined as follows:

$$ {\Psi}^{\ast}\Psi $$
Furthermore, if F and G are two different properties, it turns out that
  • 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

$$ \int {\varphi}_m^{\ast }\ \widehat{A}\ {\varphi}_n\ d\tau \equiv \left\langle {\varphi}_m\mid \widehat{A}\mid {\varphi}_n\right\rangle \equiv \left({\varphi}_m\mid \widehat{A}\mid {\varphi}_n\right)\equiv \left\langle m\mid \widehat{A}\mid n\right\rangle \equiv {\widehat{A}}_{mn} $$
$$ \int {\varphi}_m^{\ast }\ {\varphi}_n\ \mathrm{d}\tau \equiv \left\langle {\varphi}_m\mid {\varphi}_n\right\rangle \equiv \left\langle m\mid n\right\rangle $$

In the above equations, 〈m | n〉 is the Dirac representation of the inner product between the vectors m and n. Using Dirac bra-ket 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:

$$ \mathrm{Ket}\left(\psi \right)=\mid \psi \kern0.15em \Big\rangle =\left(\begin{array}{c}a\\ {}b\end{array}\right) $$
$$ \mathrm{Bra}\left(\psi \right)=\Big\langle \kern0.15em \psi \mid =\left(a\kern0.5em b\right) $$

The inner product for m and n is as follows.

If
$$ \mathbf{m}=\mid m\kern0.15em \left\rangle =\left(\begin{array}{c}{m}_1\\ {}{m}_2\end{array}\right)\to \right\langle m\mid =\left({m}_1^{\ast}\kern0.5em {m}_2^{\ast}\right) $$
and
$$ \mathbf{n}=\mid n\kern0.15em \left\rangle =\left(\begin{array}{c}{n}_1\\ {}{n}_2\end{array}\right)\to \right\langle n\mid =\left({n}_1^{\ast}\kern0.5em {n}_2^{\ast}\right) $$

Then

$$ \left\langle m\mid n\right\rangle =\left({m}_1^{\ast}\kern0.5em {m}_2^{\ast}\right)\ \left(\begin{array}{c}{n}_1\\ {}{n}_2\end{array}\right)={m}_1^{\ast}\times {n}_1+\kern0.5em {m}_2^{\ast}\times {n}_2 $$

Let us illustrate this with an example. Assume that

$$ \mid m\left\rangle =\left(\begin{array}{c}1+i\\ {}2-i\end{array}\right)\to \right\langle m\mid =\left(1-i\kern0.5em 2+i\right) $$
$$ \mid n\left\rangle =\left(\begin{array}{c}2-i\\ {}1-2i\end{array}\right)\to \right\langle n\mid =\left(2+i\kern0.5em 1+2i\right) $$

Then

$$ \left\langle m\mid n\right\rangle =\left(1-i\kern0.5em 2+i\right)\ \left(\begin{array}{c}2-i\\ {}1-2i\end{array}\right) $$
$$ =\left(1-i\right)\times \left(2-i\right)+\left(2+i\right)\times \left(1-2i\right)=5-6i $$
Note that it is very easy to prove the following two equations:
  • \( {\left\langle m\mid n\right\rangle}^{\ast }=\left\langle n\mid m\right\rangle \)

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

These are verified as follows:

$$ {\displaystyle \begin{array}{c}\left\langle n\mid m\right\rangle =\left(2+i\kern0.5em 1+2i\right)\ \left(\begin{array}{c}1+i\\ {}2-i\end{array}\right)=\\ {}\left(2+i\right)\times \left(1+i\right)+\left(1+2i\right)\times \left(2-i\right)=5+6i\\ {}{\left(5-6i\right)}^{\ast }=\left(5+6i\right)\end{array}} $$

And

$$ {\displaystyle \begin{array}{c}\left\langle m|m\right\rangle =\left(1-i\kern0.5em 2+i\right)\ \left(\begin{array}{c}1+i\\ {}2-i\end{array}\right)=\\ {}\left(1-i\right)\times \left(1+i\right)+\left(2+i\right)\times \left(2-i\right)=7\\ {}{(7)}^{\ast }=(7)\end{array}} $$

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, zero-point-energy-induced 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 i2 = −1.

  • Differential equation: an equation involving differentials or derivatives.

  • Diffraction: the spreading of waves around obstacles.

  • Eigenfunction: any non-zero 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 quantum-mechanical 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 space-time). The curvature of space-time 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 self-adjoint matrix or square matrix with complex entries that is equal to its own conjugate transpose. Thus, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th 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

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Vicente Moret-Bonillo
    • 1
  1. 1.Departamento de ComputaciónUniversidad de A CoruñaA CoruñaSpain

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