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Quantum superposition

From Wikipedia, the free encyclopedia
Principle of quantum mechanics
For broader coverage of this topic, seeSuperposition principle.

Quantum superposition of states and decoherence
Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

Quantum superposition is a fundamental principle ofquantum mechanics that states that linear combinations of solutions to theSchrödinger equation are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is alinear differential equation in time and position. More precisely, the state of a system is given by alinear combination of all theeigenfunctions of the Schrödinger equation governing that system.

An example is aqubit used inquantum information processing. A qubit state is most generally a superposition of the basis states|0⟩{\displaystyle |0\rangle } and|1⟩{\displaystyle |1\rangle }:

|Ψ⟩=c0|0⟩+c1|1⟩,{\displaystyle |\Psi \rangle =c_{0}|0\rangle +c_{1}|1\rangle ,}

where|Ψ⟩{\displaystyle |\Psi \rangle } is thequantum state of the qubit, and|0⟩{\displaystyle |0\rangle },|1⟩{\displaystyle |1\rangle } denote particular solutions to the Schrödinger equation inDirac notation weighted by the twoprobability amplitudesc0{\displaystyle c_{0}} andc1{\displaystyle c_{1}} that both are complex numbers. Here|0⟩{\displaystyle |0\rangle } corresponds to the classical 0bit, and|1⟩{\displaystyle |1\rangle } to the classical 1 bit. The probabilities of measuring the system in the|0⟩{\displaystyle |0\rangle } or|1⟩{\displaystyle |1\rangle } state are given by|c0|2{\displaystyle |c_{0}|^{2}} and|c1|2{\displaystyle |c_{1}|^{2}} respectively (see theBorn rule). Before the measurement occurs the qubit is in a superposition of both states.

The interference fringes in thedouble-slit experiment provide another example of the superposition principle.

Wave postulate

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The theory of quantum mechanics postulates that awave equation completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to belinear andhomogeneous. These conditions mean that for any two solutions of the wave equation,Ψ1{\displaystyle \Psi _{1}} andΨ2{\displaystyle \Psi _{2}}, a linear combination of those solutions also solve the wave equation:Ψ=c1Ψ1+c2Ψ2{\displaystyle \Psi =c_{1}\Psi _{1}+c_{2}\Psi _{2}}for arbitrary complex coefficientsc1{\displaystyle c_{1}} andc2{\displaystyle c_{2}}.[1]: 61  If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.

Transformation

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The quantum wave equation can be solved using functions of position,Ψ(r→){\displaystyle \Psi ({\vec {r}})}, or using functions of momentum,Φ(p→){\displaystyle \Phi ({\vec {p}})} and consequently the superposition of momentum functions are also solutions:Φ(p→)=d1Φ1(p→)+d2Φ2(p→){\displaystyle \Phi ({\vec {p}})=d_{1}\Phi _{1}({\vec {p}})+d_{2}\Phi _{2}({\vec {p}})}The position and momentum solutions are related by alinear transformation, aFourier transformation. This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.[1]: 244 

Generalization to basis states

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Other transformations express a quantum solution as a superposition ofeigenvectors, each corresponding to a possible result of a measurement on the quantum system. An eigenvectorψi{\displaystyle \psi _{i}} for a mathematical operator,A^{\displaystyle {\hat {A}}}, has the equationA^ψi=λiψi{\displaystyle {\hat {A}}\psi _{i}=\lambda _{i}\psi _{i}}whereλi{\displaystyle \lambda _{i}} is one possible measured quantum value for the observableA{\displaystyle A}. A superposition of these eigenvectors can represent any solution:Ψ=∑naiψi.{\displaystyle \Psi =\sum _{n}a_{i}\psi _{i}.}The states likeψi{\displaystyle \psi _{i}} are called basis states.

Compact notation for superpositions

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Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in theDirac bra-ket notation:[1]: 245 |v⟩=d1|1⟩+d2|2⟩{\displaystyle |v\rangle =d_{1}|1\rangle +d_{2}|2\rangle }This approach is especially effective for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics, and superposition of basis states is a fundamental tool in quantum mechanics.

Consequences

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Paul Dirac described the superposition principle as follows:

The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states,A andB, such that there exists an observation which, when made on the system in stateA, is certain to lead to one particular result,a say, and when made on the system in stateB 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 sometimesa and sometimesb, according to a probability law depending on the relative weights ofA andB in the superposition process. It will never be different from botha andb [i.e., eithera orb].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.[2]

Anton Zeilinger, referring to the prototypical example of thedouble-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 isthe essential criterion for quantum interference to appear.[3]

Theory

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General formalism

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Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:

|α⟩=∑ncn|n⟩,{\displaystyle |\alpha \rangle =\sum _{n}c_{n}|n\rangle ,}

where|n⟩{\displaystyle |n\rangle } are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates,|x⟩{\displaystyle |x\rangle }:

|α⟩=∫dx′|x′⟩⟨x′|α⟩,{\displaystyle |\alpha \rangle =\int dx'|x'\rangle \langle x'|\alpha \rangle ,}

whereϕα(x)=⟨x|α⟩{\displaystyle \phi _{\alpha }(x)=\langle x|\alpha \rangle } is the projection of the state into the|x⟩{\displaystyle |x\rangle } basis and is called the wave function of the particle. In both instances we notice that|α⟩{\displaystyle |\alpha \rangle } can be expanded as a superposition of an infinite number of basis states.

Example

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Given the Schrödinger equation

H^|n⟩=En|n⟩,{\displaystyle {\hat {H}}|n\rangle =E_{n}|n\rangle ,}

where|n⟩{\displaystyle |n\rangle } indexes the set of eigenstates of the Hamiltonian with energy eigenvaluesEn,{\displaystyle E_{n},} we see immediately that

H^(|n⟩+|n′⟩)=En|n⟩+En′|n′⟩,{\displaystyle {\hat {H}}{\big (}|n\rangle +|n'\rangle {\big )}=E_{n}|n\rangle +E_{n'}|n'\rangle ,}

where

|Ψ⟩=|n⟩+|n′⟩{\displaystyle |\Psi \rangle =|n\rangle +|n'\rangle }

is a solution of the Schrödinger equation but is not generally an eigenstate becauseEn{\displaystyle E_{n}} andEn′{\displaystyle E_{n'}} are not generally equal. We say that|Ψ⟩{\displaystyle |\Psi \rangle } is made up of a superposition of energy eigenstates. Now consider the more concrete case of anelectron that has eitherspin up or down. We now index the eigenstates with thespinors in thez^{\displaystyle {\hat {z}}} basis:

|Ψ⟩=c1|↑⟩+c2|↓⟩,{\displaystyle |\Psi \rangle =c_{1}|{\uparrow }\rangle +c_{2}|{\downarrow }\rangle ,}

where|↑⟩{\displaystyle |{\uparrow }\rangle } and|↓⟩{\displaystyle |{\downarrow }\rangle } 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:

P(|↑⟩)=|c1|2,{\displaystyle P{\big (}|{\uparrow }\rangle {\big )}=|c_{1}|^{2},}
P(|↓⟩)=|c2|2,{\displaystyle P{\big (}|{\downarrow }\rangle {\big )}=|c_{2}|^{2},}
Ptotal=P(|↑⟩)+P(|↓⟩)=|c1|2+|c2|2=1,{\displaystyle P_{\text{total}}=P{\big (}|{\uparrow }\rangle {\big )}+P{\big (}|{\downarrow }\rangle {\big )}=|c_{1}|^{2}+|c_{2}|^{2}=1,}

where the probability of finding the particle with either spin up or down is normalized to 1. Notice thatc1{\displaystyle c_{1}} andc2{\displaystyle c_{2}} are complex numbers, so that

|Ψ⟩=35i|↑⟩+45|↓⟩.{\displaystyle |\Psi \rangle ={\frac {3}{5}}i|{\uparrow }\rangle +{\frac {4}{5}}|{\downarrow }\rangle .}

is an example of an allowed state. We now get

P(|↑⟩)=|3i5|2=925,{\displaystyle P{\big (}|{\uparrow }\rangle {\big )}=\left|{\frac {3i}{5}}\right|^{2}={\frac {9}{25}},}
P(|↓⟩)=|45|2=1625,{\displaystyle P{\big (}|{\downarrow }\rangle {\big )}=\left|{\frac {4}{5}}\right|^{2}={\frac {16}{25}},}
Ptotal=P(|↑⟩)+P(|↓⟩)=925+1625=1.{\displaystyle P_{\text{total}}=P{\big (}|{\uparrow }\rangle {\big )}+P{\big (}|{\downarrow }\rangle {\big )}={\frac {9}{25}}+{\frac {16}{25}}=1.}

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

Ψ=ψ+(x)⊗|↑⟩+ψ−(x)⊗|↓⟩,{\displaystyle \Psi =\psi _{+}(x)\otimes |{\uparrow }\rangle +\psi _{-}(x)\otimes |{\downarrow }\rangle ,}

where we have a general stateΨ{\displaystyle \Psi } is the sum of thetensor products of the position space wave functions and spinors.

Experiments

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Successful experiments involving superpositions ofrelatively large (by the standards of quantum physics) objects have been performed.

  • Apiezoelectric "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.[8]
  • Recent research indicates thatchlorophyll withinplants 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.[9][10]

In quantum computers

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Inquantum computers, aqubit is the analog of the classical informationbit, but rather than having one of two distinct values, qubits are a superposition of two values.[11]: 13  Controlling this superposition qubits is a central challenge in quantum computation. The superposition needs to be robust to unintended interactions and yet interaction are needed for computing with qubits. Qubit systems likenuclear spins with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.[11]: 278 

See also

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  • Eigenstate – Mathematical entity to describe the probability of each possible measurement on a systemPages displaying short descriptions of redirect targets
  • Mach–Zehnder interferometer – Device to determine relative phase shift
  • Penrose interpretation – Interpretation of quantum mechanics
  • Pure qubit state – Basic unit of quantum informationPages displaying short descriptions of redirect targets
  • Quantum computation – Computer hardware technology that uses quantum mechanicsPages displaying short descriptions of redirect targets
  • Schrödinger's cat – Thought experiment in quantum mechanics
  • Superposition principle – Fundamental physics principle stating that physical solutions of linear systems are linear
  • Wave packet – Short "burst" or "envelope" of restricted wave action that travels as a unit

References

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  1. ^abcMessiah, Albert (1976).Quantum mechanics. 1 (2 ed.). Amsterdam: North-Holland.ISBN 978-0-471-59766-7.
  2. ^P.A.M. Dirac (1947).The Principles of Quantum Mechanics (2nd ed.). Clarendon Press. p. 12.
  3. ^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.
  4. ^Monroe, C.; Meekhof, D. M.; King, B. E.; Wineland, D. J. (24 May 1996)."A "Schrödinger Cat" Superposition State of an Atom".Science.272 (5265):1131–1136.doi:10.1126/science.272.5265.1131.ISSN 0036-8075.
  5. ^"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)
  6. ^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.
  7. ^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
  8. ^O’Connell, A. D.; Hofheinz, M.; Ansmann, M.; Bialczak, Radoslaw C.; Lenander, M.; Lucero, Erik; Neeley, M.; Sank, D.; Wang, H.; Weides, M.; Wenner, J.; Martinis, John M.; Cleland, A. N. (April 2010)."Quantum ground state and single-phonon control of a mechanical resonator".Nature.464 (7289):697–703.doi:10.1038/nature08967.ISSN 0028-0836.
  9. ^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.
  10. ^Moyer, Michael (September 2009)."Quantum Entanglement, Photosynthesis and Better Solar Cells".Scientific American. Retrieved12 May 2010.
  11. ^abNielsen, Michael A.;Chuang, Isaac (2010).Quantum Computation and Quantum Information. Cambridge:Cambridge University Press.ISBN 978-1-10700-217-3.OCLC 43641333.

Further reading

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Interpretations
Experiments
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