Quantum
Einstein, Bohr, And The Great Debate About the Nature of Reality
In his quest for a new mechanics for the quantised world of the atom, Heisenberg concentrated on the frequencies and relative intensities of the spectral lines produced when an electron instantaneously jumped from one energy level to another. He had no other choice; it was the only available data about what was happening inside an atom. Despite the imagery conjured up by all the talk of quantum jumps and leaps, an electron did not 'jump' through space as it moved between energy levels like a boy jumping off a wall onto the pavement below. It was simply in one place and an instant later it popped up in another without being anywhere in between. Heisenberg accepted that all observables, or anything connected with them, were associated with the mystery and magic of the quantum jump of an electron between two energy levels. Lost forever was the picturesque miniature solar system in which each electron orbited a nuclear sun.
On the pollen-free haven of Helgoland, Heisenberg devised a method of book-keeping to track all possible electron jumps, or transitions, that could occur between the different energy levels of hydrogen. The only way he could think of recording each observable quantity, associated with a unique pair of energy levels, was to use an array: $$(v_{ij})_{m\times n}$$ This was the array for the entire set of possible frequencies of the spectral lines that could theoretically be emitted by an electron when it jumps between two different energy levels. If an electron quantum jumps from the energy level $E_2$ to the lower energy level $E_1$, a spectral line is emitted with a frequency designated by $v_{21}$ in the array. The spectral line of frequency $v_{12}$ would only be found in the absorption spectrum, since it is associated with an electron in energy level $E_1$ absorbing a quantum of energy sufficient to jump to energy level $E_2$. A spectral line of frequency vmn would be emitted when an electron jumps between any two levels whose energies are $E_m$ and $E_n$, where $m$ is greater than $n$. Not all the frequencies $v_{mn}$ are exactly observed. For example, measurement of $v_{11}$ is impossible, since it would be the frequency of the spectral line emitted in a 'transition' from energy level $E_1$ to energy level $E_1$ – a physical impossibility. Hence $v_{11}$ is zero, as are all potential frequencies when $m=n$. The collection of all non-zero frequencies, $v_{mn}$, would be the lines actually present in the emission spectrum of a particular element.
Another array could be formed from the calculation of transition rates between the various energy levels. If the probability for a particular transition, $a_{mn}$, from energy level $E_m$ to $E_n$, is high, then the transition is more likely than one with a lower probability. The resulting spectral line with frequency $v_{mn}$ would be more intense than for the less probable transition. Heisenberg realised that the transition probabilities $a_{mn}$ and the frequencies $v_{mn}$ could, after some deft theoretical manipulation, lead to a quantum counterpart for each observable quantity known in Newtonian mechanics such as position and momentum.
Schrödinger finally proposed that the wave function of an electron, for example, was intimately connected to the cloud-like distribution of its electric charge as it traveled through space. In wave mechanics the wave function was not a quantity that could be directly measured because it was what mathematicians call a complex number. 4+3i is one example of such a number, and it consists of two parts: one 'real' and the other 'imaginary'. 4 is an ordinary number and is the 'real' part of the complex number 4+3i. The 'imaginary' part, 3i, has no physical meaning because i is the square root of -1. The square root of a number is just another number that multiplied by itself will give the original number. The square root of 4 is 2 since 2×2 equals 4. There is no number that multiplied by itself equals -1. While 1×1=1, –1×–1 is also equal to 1, since by the laws of algebra, a minus times a minus generates a plus.
The wave function was unobservable; it was something intangible that could not be measured. However, the square of a complex number gives a real number that is associated with something that can actually be measured in the laboratory. The square of 4+3i is 25. Schrödinger believed that the square of the wave function of an electron, $|\Psi(x,t)|^2$ was a measure of the smeared-out density of electric charge at location x at time t.
As part of his interpretation of the wave function, Schrödinger introduced the concept of a 'wave packet' to represent the electron as he challenged the very idea that particles existed. He argued that an electron only 'appeared' to be particle-like but was not actually a particle, despite the overwhelming experimental evidence in favour of it being so. Schrödinger believed that a particle-like electron was an illusion. In reality there were only waves. Any manifestation of a particle electron was due to a group of matter waves being superimposed into a wave packet. An electron in motion would then be nothing more than a wave packet that moved like a pulse sent, with a flick of the wrist, travelling down the length of a taut rope tied at one end and held at the other. A wave packet that gave the appearance of a particle required a collection of waves of different wavelengths that interfered with one another in such a way that they cancelled each other out beyond the wave packet.
If giving up particles and reducing everything to waves rid physics of discontinuity and quantum jumps, then for Schrödinger it was a price worth paying. However, his interpretation soon ran into difficulties as it failed to make physical sense. Firstly, the wave packet representation of the electron began to unravel when it was discovered that the constituent waves would spread out across space to such a degree that they would have to travel faster than the speed of light if they were to be connected with the detection of a particle-like electron in an experiment.
Try as he might, there was no way for Schrödinger to prevent this dispersal of the wave packet. Since it was made up of waves that varied in wavelength and frequency, as the wave packet traveled through space it would soon begin to spread out as individual waves moved at different velocities. An almost instantaneous coming together, a localisation at one point in space, would have to take place every time an electron was detected as a particle. Secondly, when attempts were made to apply the wave equation to helium and other atoms, Schrödinger's vision of the reality that lay beneath his mathematics disappeared into an abstract, multi-dimensional space that was impossible to visualize.
Nor could Schrödinger's interpretation account for the photoelectric and Compton effects. There were unanswered questions: how could a wave packet possess electric charge? Could wave mechanics incorporate quantum spin? If Schrödinger's wave function did not represent real waves in everyday three-dimensional space, then what were they? It was Max Born who provided the answer.
The wave function itself has no physical reality; it exists in the mysterious, ghost-like realm of the possible. It deals with abstract possibilities, like all the angles by which an electron could be scattered following a collision with an atom. There is a real world of difference between the possible and the probable. Born argued that the square of the wave function, a real rather than a complex number, inhabits the world of the probable. Squaring the wave function, for example, does not give the actual position of an electron, only the probability, the odds that it will found here rather than there. For example, if the value of the wave function of an electron at X is double its value at Y, then the probability of it being found at X is four times greater than the probability of finding it at Y. The electron could be found at X, Y or somewhere else.
Niels Bohr would soon argue that until an observation or measurement is made, a microphysical object like an electron does not exist anywhere. Between one measurement and the next it has no existence outside the abstract possibilities of the wave function. It is only when an observation or measurement is made that the 'wave function collapses' as one of the 'possible' states of the electron becomes the 'actual' state and the probability of all the other possibilities becomes zero.
For Born, Schrödinger's equation described a probability wave. There were no real electron waves, only abstract waves of probability. 'From the point of view of our quantum mechanics there exists no quantity which in an individual case causally determines the effect of a collision', wrote Born. And he confessed, 'I myself tend to give up determinism in the atomic world.' Yet while the 'motion of particles follows probability rules', he pointed out, 'probability itself propagates according to the law of causality'.
It took Born the time between his two papers to fully grasp that he had introduced a new kind of probability into physics. 'Quantum probability', for want of a better term, was not the classical probability of ignorance that could in theory be eliminated. It was an inherent feature of atomic reality. For example, the fact that it was impossible to predict when an individual atom would decay in a radioactive sample, amid the certainty that one would do so, was not due to a lack of knowledge but was the result of the probabilistic nature of the quantum rules that dictate radioactive decay.
According to the uncertainty principle, a measurement that yields an exact value for the momentum of a microphysical object or system excludes even the possibility of simultaneously measuring its position. The question that Einstein wanted to answer was: Does the inability to measure its exact position directly mean that the electron does not have a definite position? The Copenhagen interpretation answered that in the absence of a measurement to determine its position, the electron has no position. EPR set out to demonstrate that there are elements of physical reality, such as an electron having a definite position, that quantum mechanics cannot accommodate – and therefore, it is incomplete.
EPR attempted to clinch their argument with a thought experiment. Two particles, A and B, interact briefly and then move off in opposite directions. The uncertainty principle forbids the exact measurement, at any given instant, of both the position and the momentum of either particle. However, it does allow an exact and simultaneous measurement of the total momentum of the two particles, A and B, and the relative distance between them.
The key to the EPR thought experiment is to leave particle B undisturbed by avoiding any direct observation of it. Even if A and B are light years apart, nothing within the mathematical structure of quantum mechanics prohibits a measurement of the momentum of A yielding information about the exact momentum of B without B being disturbed in the process. When the momentum of particle A is measured exactly, it indirectly but simultaneously allows, via the law of conservation of momentum, an exact determination of the momentum of B. Therefore, according to the EPR criterion of reality, the momentum of B must be an element of physical reality. Similarly, by measuring the exact position of A, it is possible, because the physical distance separating A and B is known, to deduce the position of B without directly measuring it. Hence, EPR argue, it too must be an element of physical reality. EPR appeared to have contrived a means to establish with certainty the exact values of either the momentum or the position of B due to measurements performed on particle A, without the slightest possibility of particle B being physically disturbed.
Given their reality criterion, EPR argued that they had thus proved that both the momentum and position of particle B are 'elements of reality', that B can have simultaneously exact values of position and momentum. Since quantum mechanics via the uncertainty principle rules out any possibility of a particle simultaneously possessing both these properties, these 'elements of reality' have no counterparts in the theory. Therefore the quantum mechanical description of physical reality, EPR conclude, is incomplete.
Einstein, Podolsky and Rosen had conjured up an imaginary experiment that involved a pair of correlated particles, A and B, so far apart that it should be impossible for them to physically interact with one another. EPR argued that a measurement carried out on particle A could not physically disturb particle B. Since any measurement is performed on only one of the particles, EPR believed they could cut off Bohr's counter-attack – an act of measurement causes a 'physical disturbance'. Since the properties of the two particles are correlated, they argued that by measuring a property of particle A, such as its position, it is possible to know the corresponding property of B without disturbing it. EPR's aim was to demonstrate that particle B possessed the property independently of being measured, and since this was something that quantum mechanics failed to describe, it was therefore incomplete. Bohr countered, never so succinctly, that the pair of particles were entangled and formed a single system no matter how far apart they were. Therefore, if you measured one, then you also measured the other.
The Copenhagen interpretation requires an observer outside the universe to observe it, but since there is none – leaving God aside – the universe should never come into existence but remain forever in a superposition of many possibilities. This is the long-standing measurement problem writ large. Schrödinger's equation that describes quantum reality as a superposition of possibilities, and attaches a range of probabilities to each possibility, does not include the act of measurement. There are no observers in the mathematics of quantum mechanics. The theory says nothing about the collapse of the wave function, the sudden and discontinuous change of the state of a quantum system upon observation or measurement, when the possible becomes the actual. In Everett's many worlds interpretation there was no need for an observation or measurement to collapse the wave function, since each and every quantum possibility coexists as an actual reality in an array of parallel universes.
No comments:
Post a Comment