Quantum state space

In physics, a quantum state space is an abstract space in which different "positions" represent not literal locations, but rather quantum states of some physical system. It is the quantum analog of the phase space of classical mechanics.

Relative to Hilbert space

In quantum mechanics a state space is a separable complex Hilbert space. The dimension of this Hilbert space depends on the system we choose to describe.^[1]^[2] The different states that could come out of any particular measurement form an orthonormal basis, so any state vector in the state space can be written as a linear combination of these basis vectors. Having an nonzero component along multiple dimensions is called a superposition. In the formalism of quantum mechanics these state vectors are often written using Dirac's compact bra–ket notation.^[3]^: 165

Examples

The spin state of a silver atom in the Stern–Gerlach experiment can be represented in a two state space. The spin can be aligned with a measuring apparatus (arbitrarily called 'up') or oppositely ('down').^[4] In Dirac's notation these two states can be written as $|u\rangle ,|d\rangle$ . The space of a two spin system has four states, $|uu\rangle ,|ud\rangle ,|du\rangle ,|dd\rangle$ .

The spin state is a discrete degree of freedom; quantum state spaces can have continuous degrees of freedom. For example, a particle in one space dimension has one degree of freedom ranging from $-\infty$ to $\infty$ . In Dirac notation, the states in this space might be written as $|q\rangle$ or $|\psi \rangle$ .^[5]^: 302

Relative to 3D space

Even in the early days of quantum mechanics, the state space (or configurations as they were called at first) was understood to be essential for understanding simple quantum-mechanical problems. In 1929, Nevill Mott showed that "tendency to picture the wave as existing in ordinary three dimensional space, whereas we are really dealing with wave functions in multispace" makes analysis of simple interaction problems more difficult.^[6] Mott analyzes $\alpha$ -particle emission in a cloud chamber. The emission process is isotropic, a spherical wave in quantum mechanics, but the tracks observed are linear.

As Mott says, "it is a little difficult to picture how it is that an outgoing spherical wave can produce a straight track; we think intuitively that it should ionise atoms at random throughout space". This issue became known at the Mott problem. Mott then derives the straight track by considering correlations between the positions of the source and two representative atoms, showing that consecutive ionization results from just that state in which all three positions are co-linear. ^[7]

Relative to classical phase space

Classical mechanics for multiple objects describes their motion in terms of a list or vector of every object's coordinates and velocity. As the objects move, the values in the vector change; the set of all possible values is called a phase space.^[8]^: 88 In quantum mechanics a state space is similar, however in the state space two vectors which are scalar multiples of each other represent the same state. Furthermore, the character of values in the quantum state differ from the classical values: in the quantum case the values can only be measured statistically (by repetition over many examples) and thus do not have well defined values at every instant of time. ^[5]^: 294

References

^ McIntyre, David (2012). Quantum Mechanics: A Paradigms Approach (1st ed.). Pearson. ISBN 978-0321765796.
^ Bengtsson, Ingemar; Życzkowski, Karol (2017). Geometry of Quantum States (2nd ed.). Cambridge University Press. ISBN 978-1139207010.
^ Schiff, Leonard (1949). Quantum mechanics. McGraw-Hill.
^ Susskind, Leonard; Friedman, Art; Susskind, Leonard (2014). Quantum mechanics: the theoretical minimum; [what you need to know to start doing physics]. The theoretical minimum / Leonard Susskind and George Hrabovsky. New York, NY: Basic Books. ISBN 978-0-465-06290-4.
^ ^a ^b Messiah, Albert (1966). Quantum Mechanics. North Holland, John Wiley & Sons. ISBN 0486409244.
^ "The wave mechanics of ∝-Ray tracks". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 126 (800): 79–84. 1929-12-02. doi:10.1098/rspa.1929.0205. ISSN 0950-1207.
^ Figari, Rodolfo; Teta, Alessandro (2013). "Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929)". Archive for History of Exact Sciences. 67 (2): 215–234. arXiv:1209.2665. doi:10.1007/s00407-012-0111-z. ISSN 0003-9519. S2CID 253891627.
^ Susskind, Leonard; Hrabovsky, George; Susskind, Leonard (2014). The theoretical minimum: what you need to know to start doing physics. The theoretical minimum / Leonard Susskind and George Hrabovsky (Paperback 1. publ ed.). New York: Basic Books. ISBN 978-0-465-07568-3.