Quantum state

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Quantum mechanics

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In quantum physics, a quantum state is a mathematical object that fully describes a quantum system. One typically imagines some experimental apparatus and procedure which "prepares" this quantum state; the mathematical object then reflects the setup of the apparatus. Quantum states can be statistically mixed, corresponding to a experiment involving a random change of the parameters. States obtained in this way are called mixed states, as opposed to pure states which cannot be described as a mixture of others. When performing a certain measurement on a quantum state, the result is in general described by a probability distribution, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. However, unlike in classical mechanics, the result of a measurement on even a pure quantum state is only determined probabilistically. This reflects a core difference between classical and quantum physics.

Mathematically, a pure quantum state is typically represented by a vector in a Hilbert space. In physics, bra-ket notation is often used to denote such vectors. Linear combinations (superpositions) of vectors can describe interference phenomena. Mixed quantum states are described by density matrices.

In a more general mathematical context, quantum states can be understood as positive normalized linear functionals on a C* algebra; see GNS construction.

1 Conceptual description
2 Formalism in quantum physics
3 Mathematical formulation
4 Notes
5 See also
6 Further reading

The state of a physical system is a complete description of the parameters of the experiment. To understand this rather abstract notion, it is useful to first explore it in an example from classical mechanics.

Consider an experiment with a (non-quantum) particle of mass $m = 1$ which moves freely, and without friction, in one spatial direction.

We start the experiment at time $t = 0$ by pushing the particle with some speed into some direction. Doing this, we determine the initial position $q$ and the initial momentum^[1] $p$ of the particle. These initial conditions are what characterizes the state $σ$ of the system, formally denoted as $σ = (p, q)$ . We say that we prepare the state of the system by fixing its initial conditions.

At a later time $t > 0$ , we conduct measurements on the particle. The measurements we can perform on this simple system are essentially its position $Q (t)$ at time $t$ , its momentum $P (t)$ , and combinations of these. Here $P (t)$ and $Q (t)$ refer to the measurable quantities (observables) of the system as such, not the specific results they produce in a certain run of the experiment.

However, knowing the state $σ$ of the system, we can compute the value of the observables in the specific state, i.e., the results that our measurements will produce, depending on $p$ and $q$ . We denote these values as $\langle P(t) \rangle _\sigma$ and $\langle Q(t) \rangle _\sigma$ . In our simple example, it is well known that the particle moves with constant velocity; therefore,

$\langle P(t) \rangle _\sigma = p, \quad \langle Q(t) \rangle _\sigma = pt+q.$

Now suppose that we start the particle with a random initial position and momentum. (For argument's sake, we may suppose that the particle is pushed away at $t = 0$ by some apparatus which is controlled by a random number generator.) The state $σ$ of the system is now not described by two numbers $p$ and $q$ , but rather by two probability distributions. The observables $P (t)$ and $Q (t)$ will produce random results now; they become random variables, and their values in a single measurement cannot be predicted. However, if we repeat the experiment sufficiently often, always preparing the same state $σ$ , we can predict the expectation value of the observables (their statistical mean) in the state $σ$ . The expectation value of $P (t)$ is again denoted by $\langle P(t) \rangle _\sigma$ , etc.

These "statistical" states of the system are called mixed states, as opposed to the pure states $σ = (p, q)$ discussed further above. Abstractly, mixed states arise as a statistical mixture of pure states.

Probability densities for the electron of a hydrogen atom in different quantum states

In quantum systems, the conceptual distinction between observables and states persists just as described above. The state $σ$ of the system is fixed by the way the physicist prepares his experiment (e.g., how he adjusts his particle source). As above, there is a distinction between pure states and mixed states, the latter being statistical mixtures of the former. However, some important differences arise in comparison with classical mechanics.

In quantum theory, even pure states show statistical behaviour. Regardless of how carefully we prepare the state $ρ$ of the system, measurement results are not repeatable in general, and we must understand the expectation value $\langle A \rangle _\sigma$ of an observable $A$ as a statistical mean. It is this mean that is predicted by physical theories.

For any fixed observable $A$ , it is generally possible to prepare a pure state $σ A$ such that $A$ has a fixed value in this state: If we repeat the experiment several times, each time measuring $A$ , we will always obtain the same measurement result, without any random behaviour. Such pure states $σ A$ are called eigenstates of $A$ .

However, it is generally impossible to prepare a simultaneous eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement $Q (t)$ and the momentum measurement $P (t)$ (at the same time $t$ ) produce "sharp" results; at least one of them will exhibit random behaviour.^[2] This is the content of the Heisenberg uncertainty relation.

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system changes its state. More precisely: After measuring an observable $A$ , the system will be in an eigenstate of $A$ . This expresses a kind of logical consistency: If we measure $A$ twice in the same run of the experiment, the measurements being directly consecutive in time, then they will produce the same results. This has some strange consequences however:

Consider two observables, $A$ and $B$ , where $A$ corresponds to a measurement earlier in time than $B$ .^[3] Suppose that the system is in an eigenstate of $B$ . If we measure only $B$ , we will not notice statistical behaviour. If we measure first $A$ and then $B$ in the same run of the experiment, the system will transfer to an eigenstate of $A$ after the first measurement, and we will generally notice that the results of $B$ are statistical. Thus, quantum mechanical measurements influence one another, and it is important in which order they are performed.

Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow to distinguish between quantum theory and alternative classical (non-quantum) models.

In the discussion above, we have taken the observables $P (t)$ , $Q (t)$ to be dependent on time, while the state $σ$ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. Conceptually (and mathematically), both approaches are equivalent; choosing one of them is a matter of convention.

Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory.

Paul Dirac invented a powerful and intuitive notation to describe quantum states, known as bra-ket notation. For instance, one can refer to an |excited atom> or to $|\!\!\uparrow\rangle$ for a spin-up particle, hiding the underlying complexity of the mathematical description, which is revealed when the state is projected onto a coordinate basis. For instance, the simple notation |1s> describes the first hydrogen atom bound state, but becomes a complicated function in terms of Laguerre polynomials and spherical harmonics when projected onto the basis of position vectors |r>. The resulting expression Ψ(r)=<r|1s>, which is known as the wave function, is a special representation of the quantum state, namely, its projection into position space. Other representations, such as projection into momentum space, are possible. The various representations are simply different expressions of a single physical quantum state.

Any quantum state $|\psi\rangle$ can be expressed in terms of a sum of basis states (also called basis kets) $|k_i\rangle$ in the form

$| \psi \rangle = \sum_i c_i | k_i \rangle$

where $c i$ are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, $\left | c_i \right | ^2$ is the probability of a measurement in terms of the basis states yielding the state $|k_i\rangle$ . The normalization condition mandates that the total sum of probabilities is equal to one,

$\sum_i \left | c_i \right | ^2 = 1.$

The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state $|n\rangle$ has an energy $E_n = \hbar \omega \left(n + {\begin{matrix}\frac{1}{2}\end{matrix}}\right)$ . The set of basis states can be extracted using a construction operator $\hat{a}^{\dagger}$ and a destruction operator $\hat{a}$ in what is called the ladder operator method.

If a quantum mechanical state $|\psi\rangle$ can be reached by more than one path, then $|\psi\rangle$ is said to be a linear superposition of states. In the case of two paths, if the states after passing through path $α$ and path $β$ are

$|\alpha\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle,$ and

$|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix} |1\rangle,$

then $|\psi\rangle$ is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields

$|\psi\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\alpha\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|\beta\rangle = \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) + \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}(\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|0\rangle - \begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}|1\rangle) = |0\rangle.$

Note that in the states $|\alpha\rangle$ and $|\beta\rangle,$ the two states $|0\rangle$ and $|1\rangle$ each have a probability of $\begin{matrix}\frac{1}{2}\end{matrix},$ as obtained by the absolute square of the probability amplitudes, which are $\begin{matrix}\frac{1}{\sqrt{2}}\end{matrix}$ and $\begin{matrix}\pm\frac{1}{\sqrt{2}}\end{matrix}.$ In a superposition, it is the probability amplitudes which add, and not the probabilities themselves. The pattern which results from a superposition is often called an interference pattern. In the above case, $|0\rangle$ is said to constructively interfere, and $|1\rangle$ is said to destructively interfere.

For more about superposition of states, see the double-slit experiment.

A pure quantum state is a state which can be described by a single ket vector, or as a sum of basis states. A mixed quantum state is a statistical distribution of pure states.

The expectation value $\langle a \rangle$ of a measurement $A$ on a pure quantum state is given by

where $|\alpha_i\rangle$ are basis kets for the operator $A$ , and $P (α i)$ is the probability of $| \psi \rangle$ being measured in state $|\alpha_i\rangle.$

In order to describe a statistical distribution of pure states, or mixed state, the density matrix (or density operator), $ρ,$ is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as

$\rho = \sum_s p_s | \psi_s \rangle \langle \psi_s |$

where $p s$ is the fraction of each ensemble in pure state $|\psi_s\rangle.$ The ensemble average of a measurement $A$ on a mixed state is given by

$\left [ A \right ] = \langle \overline{A} \rangle = \sum_s p_s \langle \psi_s | A | \psi_s \rangle = \sum_s \sum_i p_s a_i | \langle \alpha_i | \psi_s \rangle |^2 = tr(\rho A)$

where it is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average over the ensemble of pure states.

The von Neumann entropy of a pure state is 0.

For a mathematical discussion on states as functionals, see Gelfand-Naimark-Segal construction. There, the same objects are described in a C*-algebraic context.

^ If you are not familiar with the concept of momentum, think of it as being the velocity of the particle. That is fully justified in this context.
^ To avoid misunderstandings: Here we mean that $Q (t)$ and $P (t)$ are measured in the same state, but not in the same run of the experiment.)
^ For concreteness' sake, you may suppose that $A = Q (t 1)$ and $B = P (t 2)$ in the above example, with $t 2 > t 1 > 0$ .

The concept of quantum states, in particular the content of the section Formalism in quantum physics above, is covered in most standard textbooks on quantum mechanics.

For a discussion of conceptual aspects and a comparison with classical states, see:

Isham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. ISBN 978-1860940019.

For a more detailed coverage of mathematical aspects, see:

Bratteli, Ola; Robinson, Derek W (1987). Operator Algebras and Quantum Statistical Mechanics 1. Springer. 2nd edition. ISBN 978-3540170938. In particular, see Sec. 2.3.

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