Quantum Science Philippines
Quantum Science Philippines

Max Born’s Statistical Interpretation

Liza Marie T. Dangkulos

This article contains a summary of Max Born’s Nobel lecture entitled, “The statistical interpretation of quantum mechanics”.

In 1926, shortly after the formulation of the Schrodinger’s equation, Max Born studied the scattering of a beam of electrons and was led to his interpretation of the wave function in the said equation.

Born’s statistical interpretation states that:
The probability of finding an electron, described by the wave function, Ψ (x,t), in the region lying between x and x+dx is given by:

formula2

where

formula3 is the complex square or Ψ*Ψ[1]

He, therefore, introduced the statistical point of view into modern physics.[2] For this invaluable contribution in the field of quantum mechanics, Born was awarded the Nobel Prize in Physics in 1954.

During his Nobel lecture, Born accounted the developments in the field of quantum mechanics that led him to his statistical interpretation. He mentioned that in 1925, he and Werner Heisenberg formulated the matrix mechanics representation of quantum mechanics. Wolfgang Pauli consequently calculated the stationary energy values of hydrogen atom by means of the matrix method and from this moment onwards, there could no longer be any doubt about the correctness of the theory.

In 1926, Louis de Broglie formulated the de Broglie hypothesis claiming that all matter, has a wave-like nature. He related wavelength (denoted by λ) and momentum (denoted by p) as:

λ=h/p

where h is the Planck’s constant

Schrodinger, following de Broglie’s wave-particle duality theory of matter, constructed his famous equation that describes how the quantum state of a physical system changes in time. This equation can be mathematically transformed into matrix mechanics.

Not long after, Born developed his statistical interpretation. Not only was it developed from Schrodinger’s equation but from Einstein’s idea as well. Einstein interpreted the square of the optical wave amplitudes as the probability density for the occurrence of photons. For Born, this concept could be carried over to the Ψ-function.  Ψ*Ψ represents the probability density for electrons.

Furthermore, Born also emphasized that the indeterministic statistical interpretation should be accepted despite the strong oppositions of some respected physicists like Erwin Schrodinger, Louis de Broglie and Albert Einstein. He believed that Heisenberg’s uncertainty principle contributed to the swift acceptance of the statistical interpretation of the Ψ-function.

Uncertainty principle states that certain pairs of physical properties cannot both be known to arbitrary precision. Its meaning, according to Heisenberg, is that it is impossible to determine simultaneously both the position and velocity of an electron or any other particle with any great degree of accuracy. With this, Born had this to say, “Can absolute prediction really be made for all the time on the basis of the classical equations of motion?”

Towards the end of his lecture, Born made these two statements: “Classical physics cannot be used as an objection to the essentially indeterministic statistical interpretation of quantum mechanics” and that “I am emphatically in favour of the retention of the particle idea.”[3]

Through his statistical interpretation, Max Born showed that the solution of the Schrodinger equation has a physical significance.

[1] Stephen Gasiorowics. Quantum Physics, 3rd ed. (John Wiley and Sons, Inc., 2003) p. 28.
[2] Walter Greiner. Quantum Mechanics: An Introduction, 4th ed. (Springer-Verlag, Berlin) p.65
[3] Max Born. The Statistical Interpretation of Quantum Mechanics. Nobel Lecture, 1954.

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Chip-based quantum computer using trap technique

John Paul J. Aseniero

Computers today are now fundamental part of people’s lives. It is used in a lot of applications such as in business, communication, security systems, sciences and etc. Developing fast classical computer has come to its fundamental limitation and aiming this type of computer would rely on making the device smaller to make chips’ transistor switch faster. However, when they begin to approach 10 nanometers, electrons will start revealing their quantum nature and very strange things will happen. When transistors reach those infinitesimal dimensions and electrons start showing their true colors, this will be the start of vast new frontiers for computing which is based on quantum computers.

Finding something to act as quantum bit or qubit whose quantum state can be read and manipulated is the first thing to remember in building a quantum computer. However, quantum state is a frail thing for it can easily be changed by just a fluctuation of magnetic field or a strong-willed photon interaction. By then, two physicists from Austria’s University of Innsbruck, Juan Ignacio Cirac and Peter Zoller, theorized that a string of ions held fast in a vacuum by an electromagnetic field and cooled to within a few thousandths of a degree above absolute zero could act as stable qubits and form the basis of a quantum computer. There are also research group in NIST that had lot of experience in trapping and cooling ions from their work of atomic clock and one example of their work is trapping beryllium ion as qubit to perform logic operations which is the main key in running a quantum computer.

Even before, physicists have come up with at least half a dozen ways to do quantum computation. This includes using atomic nuclei in organic compounds as qubits and manipulating electrons within superconducting loop. However, it’s hard to handle more than a dozen of qubits which will never lead to an efficient quantum computer that requires hundreds if not thousands. It’s hard to create a full scale ion trap big enough to accommodate that many qubits. Therefore, the only way to build quantum computer is to build the equivalent of quantum integrated circuits. Trap technique is the best way to create these quantum transistors that work the same way like to shrink them down enough and put many of them of the same piece of semiconductor.

Quantum computers could one day replace silicon chips, just like the transistor once replaced the vacuum tube. But for now, the technology required to develop such a quantum computer is beyond our reach. Most research in quantum computing is still very theoretical. There is difficulty in some aspect of building this quantum computer because an equivalent of very large scale integration would require handling the control circuitry just to move the ions around. Five thousand ions would need many dozens of lasers for cooling, detection, and gate operations which should be precisely controlled in coordination with the ions’ motion in the trap. Therefore, this needs a great deal of infrastructure, including a powerful classical computer, to run a useful quantum computer. The most advanced quantum computers have not gone beyond manipulating more than 16 qubits, meaning that they are a far cry from practical application. However, the potential remains that quantum computers one day could perform, quickly and easily, calculations that are incredibly time-consuming on conventional computers. But there is still hope since scientists are running today and plan to run in the near future will almost certainly lead to insights that could make full-scale quantum computing.

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Exclusion Principle of Wolfgang Ernst Pauli

Rommel J. Jagus

Wolfgang Ernst Pauli was one of the great contributors of quantum mechanics. He was an Austrian-born Swiss physicist and a Nobel laureate. He was born in Vienna Austria – Hungary on April 25 of year 1900. In 1918, he finished his early education in Vienna. He received his Ph.D. in July 1921 for his thesis on the quantum theory of ionized molecular hydrogen under his doctoral adviser Arnold Sommerfeld.

Pauli was influenced by Bohr’s lectures in understanding the concept of atomic model. The question, as to why all electrons for an atom in the ground state were not bound in the innermost shell, that Bohr tried to answer has no convincing explanations during his lectures in Gottingen. Pauli’s eager to answer the question also led to answer another phenomenon. He was in Copenhagen when he made a serious effort to explain the formation of douplet-spectra of the alkali metals spectra for which they called the Anomalous Zeeman Effect. This type of splitting exhibited of the spectral lines in a magnetic field is different from the normal triplet by normal Zeeman Effect. The reason of Bohr was that a non-vanishing angular momentum of the atomic core was supposed cause of this douplet structure. Pauli argues with this reason which he rejected and instead of it he proposed a new quantum theoretic property of the electron, which he called a “two-valuedness not describable classically”. The idea of electron spin were introduced by Uhlenbeck and Goudsmit, which made Pauli understand the anomalous Zeeman effect by simply assuming a spin quantum number of one electron is equal 1/2. Since then, idea of exclusion principle has been closely connected with the idea of spin. The idea of spin is then become essential to quantum-mechanical property of electron and to the field of quantum mechanics. Hence, exclusion principle states that no two electrons can occupy the same quantum or energy state of an atom simultaneously.

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BELL’S INEQUALITY: A Distinction between Local Reality and Quantum Mechanics Theory

Marichu M. Tompong-Miscala

Local realism is the idea that an object has definite properties or assumes a definite state without being affected by the act of measurement [1]. This belief is especially contrary to the probabilistic ( not deterministic ) characteristic of quantum mechanics.

In 1965, John S. Bell had proposed a mathematical proof or designed several experiments that would test the consistency of quantum mechanics, and hence the inconsistency of the local reality. This has brought the idea that quantum mechanics is an incomplete theory. That is, physical properties such as momentum and position were absolute values and that when they exist, whether they were measured or not, an inequality ( Bell’s inequality), would then be satisfied. But later on this theory was the subject of considerable interests and debates, and loopholes are uncovered by the much refined experiments. Moreover, several proposals for closing this loophole have been made as well, and previous investigation was reported.

Early experiments such as the experiment in the correlation measurements in the classical properties of massive entangled particles Be+ ions, were made to test Bell’s inequalities and these were subject to two primary loopholes. The first primary loophole might be termed as ‘locality’ or ‘lightcone’ loophole, in which the correlations of apparently separate events, could result from unknown subluminal ‘signals’ propagating between different regions of the apparatus. These correlations violate a form of Bell’s inequality which was obtained by a complete set of measurements. Here, the appropriate ‘Bell’s signal’ is  2.25 +/- 0.03, whereas a value maximum of 2 is the only allowable value by local realist. Similar results have also been reported for the Geneva experiment [2]. The second loophole is usually referred to as the detection loophole. Here, every experiments is assumed to have had detection efficiencies low enough to allow the possibility that the sub-ensemble of detected events agrees with quantum mechanics even though the entire ensemble satisfies Bell’s inequalities. Conversely saying, the detected events thus represent the entire ensemble; a fair-sampling hypothesis. Thus, in the presence of an ‘accurate set’ of measurements, Bell’s inequalities are violated. Meaning, as a way of eliminating the so-called “detection-loophole”, more high-detection efficiency experiments were designed that would put distinction between what local realism is, and when the quantum mechanics theory should be applied.

[1]  M.A. Rowe, D. Kielspinski, V. Meyer, C.A. Sackett, W.M. Itano, C. Monroe, and D.J. Wineland. Experimental violation of a Bell’s inequality with efficient detection. Letters to nature. Nature 791, Vol 409 (2001)

[2]  Tittel, W., Brendel J., Zbinden H. and Gisin N. Violation of Bell’s Inequalities by photons more than 10 km apart. Phys. Rev. Lett. 81, 3563-356 (1998)

___________

M.T. Miscala is an aspiring MS Physics student of Mindanao State University-Iligan Institute of Technology. Her research interests include structural acoustics, nanotechnology and high energy physics research.

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A “Schrodinger Cat” Superposition

Sandra L. Manulat

Schrodinger’s cat paradox is a classical illustration of the conflict between the existence of quantum superpositions and our real-world experience of observation and measurement”[1].

The Principle of Superposition of States in quantum mechanics has become strikingly successful at describing physical phenomena at the atomic scale.  Quantum superposition requires us to assume that between states there exists a peculiar relationship such that whenever the system is in one state we can consider it as partly in the other states.

Now, what if we extend quantum superposition to macroscopic systems? This is where the Schrodinger Cat comes to the picture.

Schrodinger Cat started out to be a thought experimenent: An unfortunate cat is placed in a quantum superposition of being dead and alive.  How was this done? simply put, a cat was placed in a box together with a single radioactive atom that has and has not decayed.  The state of the system can be represented by the entangled quantum mechanical wave function:

[eq]\Psi = \frac{|\ddot\smile\rangle |\uparrow\rangle + |\ddot\frown\rangle|\downarrow\rangle}{\sqrt{2}}[/eq]

where [eq]|\ddot\smile\rangle[/eq] and [eq]|\ddot\frown\rangle[/eq] refer to the states of a live and dead cat, and [eq]|\uparrow\rangle[/eq] and [eq]|\downarrow\rangle[/eq] refer to the internal states of an atom that has and has not radioactively decayed.  We know that if the atom has not decayed the cat is alive and dead otherwise; but of course if we open the box, we only observe a live or dead cat and not both states.

Although it is quite impossible  for a Schrodinger’s cat (SC) to exist in the macroscopic world, there is great interest in creating SC-like states in mesoscopic systems, or systems that have both microscopic and macroscopic features.  SC-like states may provide a testing ground for the controversial theory of quantum measurement and the universality of the quantum theory.

The SC-like state was created by forming a superposition of two coherent-state wave packets of a single trapped atom with a sequence of laser pulses.  Each wave packet is correlated with a particular internal state of the atom.  A [eq]^9Be^+[/eq] ion was confined in a coaxial-resonator radio frequency trap that provides the harmonic oscillator frequencies.  The ion was laser cooled to the zero point energy and then its internal (electronic) and external (motional-nearly classical) state was coherently manipulated by applying pairs of off-resonant laser beams.  The SC superposition was verified by detection of the quantum mechanical interference between the localized wave packets[1].  The downside of the experiment is when the SC state is coupled to a thermal reservoir the superposition decays exponentially into a statistical mixture, that is the lifetime of the superposition shortens.  This phenomenon is called decoherence which explains why superpositions are rare if not impossible in the macroscopic scale, and illustrates the difficulty in preparing and maintaining even mesoscopic superpositions.

[1]C. Monroe, et. al. A “Schrodinger Cat” Superposition State of an Atom

About the author:

Sandra Manulat is taking her master’s degree in physics at MSU-Iligan Institute of Technology.  She aspires to become a physicist her country could be proud of.

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