States in Quantum Mechanics

©Fernando Caracena 2013

For those who would like to e-attend a lecture on this subject, go to the link for Dr. Leonard Susskind at Stanford, about quantum mechanics. Note that the material presented here is complimentary to what Prof. Susskind presents, and his lecture is no substitute for this article, nor is this presentation a substitute for the video. The two are complimentary.

The Photon

Quantum theory began historically with the realization that the emission and the absorption of light is corpuscular. Max Planck discovered that there is a minimum unit of light energy or quantum (ε) transferred by light energy of a given frequency (ν)

ε = h ν.                                                                                           (5)

It means that when light is sufficiently attenuated it hits as discrete impacts of quanta. These discrete excitations of the radiation field suggested the notion of point particles to early researches in quantum mechanics. The early quantum theorist pictured them as small objects moving over well defined trajectories. For this reason they are often called photons. The word “particle” is often used in modern physics , but not in the sense of classical physics where “particle” means a point of mass/energy. Instead, the word “particle” in modern physics means a quantum of energy and other properties of the object in question, which interacts as a whole unit with matter over a small spatial region and over a short period of time delivering its full load of properties as a particle would. However, quantum theory does not picture a quantum as a point-like entity moving from source to impact point as a particle, but rather as a single particle state, which has mixed particle and wave properties.

Classical vs. quantum mechanical notion of a state.

In classical physics the state of a system is defined by the configuration and instantaneous motion of its constituent particles at a particular time, which are specified by the momenta and positions of the particles. The state of the classical system is described by the location of a single point in a 6-n dimensional phase space consisting of all the momentum and position coordinates, which are also functions of time. The time evolution of the system is described by the trajectory of the point, representing the whole system, through this phase space. An entirely different mathematical scheme describes the state of a system of particles in quantum mechanics.

In quantum mechanics, the dynamical variables no longer define a state directly. Instead, it is described by a complex vector (|Ψ>), which can be infinite dimensional . The dynamical variables, such as position and momentum components of the constituent particles are operators that act on the state vector and sometimes transform it in to a new state vector. The description of physical reality is thus separated into the two types of mathematical objects: state vectors, and operators. Further, the operators have the important restrictions on them that pairs of them, called conjugate variables, cannot be measured simultaneously to arbitrary precision, but are subject to Heisenberg' s Uncertainty relations. Conjugate variables are sometimes corresponding components of vectors for the each particle of a system of particles. For example, corresponding components of position and momentum are subject to the following uncertainty relations for each particle of a system, labeled by the index, i:

Δxi Δp1i ≥ ½ħ ,                                                                 (1a)

Δyi Δp2i ≥ ½ħ ,                                                                  (1b)

Δzi Δp3i ≥ ½ħ ;                                                                  (1c)


i=1,2,3,...,N for an n particle system;

ħ = h/(2π) ;                                                                      (1d)

and h is Planck' s constant

h = 6.62606957(29)×10−34 J·s .

The Heisenberg Uncertainty relations follow automatically in quantum theory from a condition that conjugate variables such as x and px fail to commute for each particle,

x p1– p1 x = i ħ                                                                            (2a)

[x, p1] = x p1 – p1 x                                                                     (2b)

[xq, pxr] = i ħ δq r                                                                         (2c)

Where δq r is the Kronecker delta, defined as

.                                               (2d)

The commutation relation between conjugate variables of each particle does not allow one to define a  the state of a quantum mechanical system terms of these variables written as numbers to arbitrary precision; instead, the conjugate variables are written as operators that have the specified commutation properties such as in (2d), which are logically connected with the Heisenberg Uncertainty Relations for the variables computed from their corresponding operators.

The connection between the commutation relations (2d) and the uncertainty relations will not be shown here, because it detracts from the main line of thought; but perhaps a later blog will address this issue.

Non-corresponding components of conjugate vector components commute with each other (as in 2d), and compute with all those of different particles.

In quantum theory, the state vector represents the most complete knowledge that we can have of any physical system. Quantum theory applies even on a macroscopic scale, where quantum effects represent no practical limit on the precision of measurement because they are minuscule. Here a classical description suffices, and the quantum description approaches the classical in the limit that quantum effects approach zero. In this respect, quantum theory represents the better theory of reality, classical physics being a good approximation on the large scale, but not on the small scale. On the subatomic scale, quantum effects dominate, so that a classical description is inadequate.

The quantum mechanical state of a system of particles corresponds to a distribution over classical configurations and momenta of the particles of the system in terms of a 6-N dimensional phase space: each quantum mechanical state of the system loosely defines a continuous distribution over a minimum volume rage of that space, which corresponds an infinity of possible classical states. Furthermore, there is in phase space over which the quantum mechanical state can be defined, although its shape varies according the precision with which one chooses to measure some of conjugate variables.

Another important difference between the classical notion of a state and the quantum mechanical notion of a state is the way in which the whole idea of measurements is handled. In classical physics it is tacitly assumed that measurements can be performed upon a system of particles without disturbing the state of the system, or that in principle, that one can solve for the impact of observation of the state of the system. No such assumption is made in quantum theory. In fact, the whole formation of quantum theory allows for an uncontrollable disturbance induced on the state of a system of particles by measurement. This is accomplished in the following way:


Aop |Ψ> = α |Ψ'>,


1) the state of the system is represented by the ke vector (|Ψ>);

2) Aop is an operator (such as matrices, or derivative operator ) that transforms the state vector |Ψ'>, in the process multiplying it by a complex number α;

3) The transformation

|Ψ> → |Ψ'>

is a rotation, because the quantum dynamics preserves the amplitude of state vectors, and they are normalized to unity.

Perhaps the greatest difference between the quantum theory and classical physics is in the level of its abstraction of the observables. In classical physics, the symbols such as E, p and r correspond directly to observables, which stand directly for measured values of energy, momentum, and position themselves. One plugs them in as numbers into a formula and obtains other numbers, which are the predictions of the theory. In this respect classical physics is a direct description of what we can see. This is not true of quantum theory. In the quantum theory, the symbols which are analogues of the classical variables are linear operators. They do not directly represent the numbers that are generated through measurements (observables); in fact, the quantum mechanical operator's acting on state vectors and evaluated by expectation operations, represent all that one can know about the variables in question. The numbers which correspond to measured values are indeed predicted by the quantum theory, but this happens indirectly. The predicted numbers are generated by taking sandwiched scalar products of operators (such as X, here written as bold, upper case characters) with a state vector


x = <Ψ| X |Ψ> .                                                                                (3a)


The state vector | Ψ > and its Hermitian adjoint, <Ψ| (its dual) ,called "bra" and "ket" vectors, respectively, are the invention of P. A. M. Dirac, an English genius who won the Nobel Prize for his famous theory of the electron published in 1928.

Although quantum theory is an order of abstraction further removed from experiment compared to classical physics, nevertheless as in the rest of physics, it remains an experimental science, because it makes predictions which can be confirmed or disproved by experiment. Note that despite the protests of early contributors to the development of quantum theory, such as Planck, deBroglie, Einstein and Schrödinger over the direction that quantum theory has been developing, no one has been able to find a better, common-sense theory for how subatomic particles behave. Physicists have been dragged screaming and kicking to accept the theory as it stands now because at the moment there is no conflict between quantum theory and experiment. The quantum world is so far removed from everyday experience. We cannot sense this level of physical reality directly, if we did, we probably would not be fit to handle well define, macroscopic world that we live in. R. P. Feynman, another clever Nobel prize winner, said "I think I can safely say that nobody understands quantum mechanics...Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there."

We cannot understand the way subatomic particles are and move, because they do not scale to our level of perception. Nevertheless, quantum theory does use some form of mathematics, which we can use to guide our intuition in what seems counter-intuitive, which leads to some understanding of the behavior at the quantum level. The most important thing about quantum theory is that it does agree very well with experiment. A better theory should not only feel better to our common-sense notions of reality, but should agree with experiments better than the current quantum theory does.

There are aspects of quantum theory, over which physicists disagree, without these disagreement's affecting the predictions of the theory. For example, you do not have to believe the Copenhagen interpretation to be able to predict the behavior of atoms, or groups of atoms. Conflicting, philosophical ideas surround physicists' notions about quantum mechanics as they go through their own calculations, which is the part that everybody can agree on. What is important, therefore, in any treatment of quantum mechanics, is to get some kind of working knowledge of the subject. Philosophical, or lay-people's treatments of quantum mechanics are likely to differ from each other quite a bit, and all you will get is opinion, but really no firm knowledge.

In order to gain a good working knowledge of quantum mechanics, it is first necessary to gain some intuitive feeling for the quantum mechanical notions of state vectors and operations. Classical theory does nor prepare us very well to be able to appreciate these concepts. Fortunately, there exists a simple phenomenon of light that can help us bridge this conceptual gap. We refer here to experiments involving the polarization states of light. We shall find that describing polarizations of light (in the next blog) leads to some of the mathematical features of quantum theory.

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