Quantum superposition is a fundamental principle of quantum mechanics that states that linear combinations of solutions to the Schrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a linear differential equation in time and position. More precisely, the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system.
An example is a qubit used in quantum information processing. A qubit state is most generally a superposition of the basis states and :
where is the quantum state of the qubit, and , denote particular solutions to the Schrödinger equation in Dirac notation weighted by the two probability amplitudes and that both are complex numbers. Here corresponds to the classical 0 bit, and to the classical 1 bit. The probabilities of measuring the system in the or state are given by and respectively (see the Born rule). Before the measurement occurs the qubit is in a superposition of both states.
The interference fringes in the double-slit experiment provide another example of the superposition principle.
Background
editPaul Dirac described the superposition principle as follows:
The general principle of superposition of quantum mechanics applies to the states [that are theoretically possible without mutual interference or contradiction] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely, any two or more states may be superposed to give a new state...
The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e., either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.[1]
Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:
"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.[2]
Theory
editGeneral formalism
editAny state can be expanded as a sum of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:
where are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, :
where is the projection of the state into the basis and is called the wave function of the particle. In both instances we notice that can be expanded as a superposition of an infinite number of basis states.
Example
editGiven the Schrödinger equation
where indexes the set of eigenstates of the Hamiltonian with energy eigenvalues we see immediately that
where
is a solution of the Schrödinger equation but is not generally an eigenstate because and are not generally equal. We say that is made up of a superposition of energy eigenstates. Now consider the more concrete case of an electron that has either spin up or down. We now index the eigenstates with the spinors in the basis:
where and denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:
where the probability of finding the particle with either spin up or down is normalized to 1. Notice that and are complex numbers, so that
is an example of an allowed state. We now get
If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:
where we have a general state is the sum of the tensor products of the position space wave functions and spinors.
Hamiltonian Evolution of General States
editAny quantum state (a single eigenstate of the Hamiltonian or a superposition of eigenstates) evolve according to the time dependent Schrödinger equation. These states, as previously stated, are required to be normalized and live in the Hilbert space (either finite or infinite dimensional depending on the basis eigenstates used to write the general state). Under time evolution the state remains normalized but the amplitudes of each particular eigenstate can change with time:
where generally the coefficients at time and are not the same. To evolve the original state to the new state the time evolution operator is introduced:
where is the infinitesimal time evolution operator that acts on a state at time and evolves it to the state . This notation means that the state started out as and evolved into after a time . The normalization condition on the state before and after time evolution means that is a unitary operator that becomes identity as and can be written as:
where is the Hamiltonian that defines the dynamics of the state. Additionally, the time evolution operator intuitively obeys a composition property:
which allows for arbitrary time evolution. Using the composition property the general, instead of infinitesimal, time evolution operator is derived[3]:
results in the Schrödinger equation for the time evolution operator. Multiplying by the original state vector, from the right:
gives the explicit time evolution of the new state and shows that this state must also satisfy the Schrödinger equation. However, we can work with just the time evolution operator depending on how the Hamiltonian evolves in time and if the Hamiltonians at different times commute with each other.
Case 1: Hamiltonian is Not Time Independent
editIf the Hamiltonian, , is time independent we can solve the previous differential equation to obtain[3]:
We can also obtain this expression by compounding an infinite number of infinitesimal transformations to generate a finite transformation:
Case 2: Hamiltonian is Time Dependent and Commutes at All Times
editIf the Hamiltonian has explicit time dependence but the Hamiltonians at different times all commute with each other we can again solve the Schrödinger equation for to obtain[3]:
an example of a Hamiltonian with this property is a magnetic field pointing in a direction whose magnitude changes but direction remains fixed.
Case 3: Hamiltonian is Time Dependent and Not Commuting at All Times
editThe final case is for a Hamiltonian that has explicit time dependence but does not commute with the Hamiltonian at a later time. In this case the Dyson Series is used and the final form the time evolution operator is[3]:
this may be familiar from time dependent perturbation theory and finds applications in scattering problems. An example of a Hamiltonian this property is one with a magnetic field whose magnitude remains the same but the direction switches from pointing along to .
Experiments and applications
editExperiments Realizing Superposition
editSuccessful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.[4]
- A "cat state" has been achieved with photons.[5]
- A beryllium ion has been trapped in a superposed state.[6]
- A double slit experiment has been performed with molecules as large as buckyballs and functionalized oligoporphyrins with up to 2000 atoms.[7][8]
- Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed[9]
- Very sensitive magnetometers have been realized using superconducting quantum interference devices (SQUIDS) that operate using quantum interference effects in superconducting circuits.
- A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.[10]
- Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.[11][12]
Proposed Experiments
edit- An experiment has been proposed, with a bacterial cell cooled to 10 mK, using an electromechanical oscillator.[14] At that temperature, all metabolism would be stopped, and the cell might behave virtually as a definite chemical species. For detection of interference, it would be necessary that the cells be supplied in large numbers as pure samples of identical and detectably recognizable virtual chemical species.
Applications of Superposition
edit- Quantum computation: quantum computers seek to increase computing speed by using qubits to parallelize computations.
Physical Interpretation
editAny measurement executed on a particle will collapse its wave function into an eigenstate of the Hamiltonian governing its time evolution. This means that a particle prepared into a general superposition of different eigenstates will, after the measurement, be in a definite eigenstate corresponding to the eigenvalue of that state read out from the measurement. For example, a valence electron in an atom that is in an equal superposition of its ground state and first excited state when measured will be in either the ground or first excited state. If an ensemble of atoms are prepared and the valence electron of each atom is placed in an equal superposition of its ground and first excited state then a measurement executed on all atoms will return approximately half the electrons in the ground and half in the first excited state. If this measurement is executed an infinite number of times on a single atom or once on an infinite ensemble of atoms then exactly half of the electrons will be in the first excited state and half will be in the ground state. This is analogous to having an infinite number of coins balanced on their edge on a table and covered by a box. When the box is removed the table shakes slightly and all of the coins collapse into either heads or tails with equal probability due to the act of measurement. As they are equally likely to land on either heads or tails exactly half should be heads and half should be tails after all have been counted.
The reason why macroscopic objects almost never exhibit quantum superposition is an active field of study and a proposed mechanism for the emergence of classical from quantum mechanics is due to quantum decoherence. Another proposed class of theories is that the fundamental time evolution equation is incomplete, and requires the addition of a Lindbladian, the reason for this addition and the form of the additional term varies from theory to theory. A popular theory is continuous spontaneous localization, where the Lindblad term is proportional to the spatial separation of the states. This too results in a quasi-classical probabilistic state.
See also
edit- Eigenstates – Mathematical entity to describe the probability of each possible measurement on a system
- Goodhart's law – Adage about statistical measures — when an attempt is made to use a statistical measure for purposes of control (directing), its statistical validity breaks down
- Mach–Zehnder interferometer – Device to determine relative phase shift
- Penrose interpretation – Interpretation of quantum mechanics
- Pure qubit state – Basic unit of quantum information
- Quantum computation – Technology that uses quantum mechanics
- Schrödinger's cat – Thought experiment in quantum mechanics
- Superposition principle – Fundamental physics principle stating that physical solutions of linear systems are linear
- Uncertainty principle – Foundational principle in quantum physics — Certain pairs of quantum measurements cannot both give precise results.
- Wave packet
References
edit- ^ P.A.M. Dirac (1947). The Principles of Quantum Mechanics (2nd ed.). Clarendon Press. p. 12.
- ^ Zeilinger A (1999). "Experiment and the foundations of quantum physics". Rev. Mod. Phys. 71 (2): S288–S297. Bibcode:1999RvMPS..71..288Z. doi:10.1103/revmodphys.71.s288.
- ^ a b c d Sakurai, J. J.; Napolitano, Jim (17 September 2020). "Modern Quantum Mechanics". Higher Education from Cambridge University Press. doi:10.1017/9781108587280. Retrieved 26 August 2024.
- ^ "What is the world's biggest Schrodinger cat?".
- ^ "Schrödinger's Cat Now Made Of Light". 27 August 2014.
- ^ C. Monroe, et al. A "Schrodinger Cat" Superposition State of an Atom Archived 7 January 2012 at the Wayback Machine
- ^ "Wave-particle duality of C60". 31 March 2012. Archived from the original on 31 March 2012.
{{cite web}}
: CS1 maint: bot: original URL status unknown (link) - ^ Nairz, Olaf. "standinglightwave".Yaakov Y. Fein; Philipp Geyer; Patrick Zwick; Filip Kiałka; Sebastian Pedalino; Marcel Mayor; Stefan Gerlich; Markus Arndt (September 2019). "Quantum superposition of molecules beyond 25 kDa". Nature Physics. 15 (12): 1242–1245. Bibcode:2019NatPh..15.1242F. doi:10.1038/s41567-019-0663-9. S2CID 203638258.
- ^ Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", Physical Chemistry Chemical Physics, 15: 14696-14700. arXiv:1310.8343
- ^ Scientific American: Macro-Weirdness: "Quantum Microphone" Puts Naked-Eye Object in 2 Places at Once: A new device tests the limits of Schrödinger's cat
- ^ Scholes, Gregory; Elisabetta Collini; Cathy Y. Wong; Krystyna E. Wilk; Paul M. G. Curmi; Paul Brumer; Gregory D. Scholes (4 February 2010). "Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature". Nature. 463 (7281): 644–647. Bibcode:2010Natur.463..644C. doi:10.1038/nature08811. PMID 20130647. S2CID 4369439.
- ^ Moyer, Michael (September 2009). "Quantum Entanglement, Photosynthesis and Better Solar Cells". Scientific American. Retrieved 12 May 2010.
- ^ "How to Create Quantum Superpositions of Living Things" Archived 11 January 2012 at the Wayback Machine>
- ^ Cartlidge, Edwin (21 September 2015). "Could 'Schrödinger's bacterium' be placed in a quantum superposition?". Physics World. IOP Publishing. Retrieved 25 August 2024.
Bibliography of cited references
edit- Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement 14 April 1928, 121: 580–590.
- Cohen-Tannoudji, C., Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, ISBN 0471164321.
- Dirac, P. A. M. (1930/1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press.
- Einstein, A. (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp. 665–688 in Schilpp, P. A. editor (1949), Albert Einstein: Philosopher-Scientist, volume II, Open Court, La Salle IL.
- Feynman, R. P., Leighton, R.B., Sands, M. (1965). The Feynman Lectures on Physics, volume 3, Addison-Wesley, Reading, MA.
- Merzbacher, E. (1961/1970). Quantum Mechanics, second edition, Wiley, New York.
- Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam.
- Wheeler, J. A.; Zurek, W.H. (1983). Quantum Theory and Measurement. Princeton NJ: Princeton University Press.
- Nielsen, Michael A.; Chuang, Isaac (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0521632358. OCLC 43641333.
- Williams, Colin P. (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.
- Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum computing for computer scientists. Cambridge University Press. ISBN 978-0-521-87996-5.