Quantum State
A Quantum State is a physical state of a quantum system.
- Example(s):
- See: Quantum Physics, Quantum Bit, Hilbert Space, Electron, Hydrogen Atom, Principal Quantum Number, David Bohm, EPR Experiment, Spin (Physics), Complex Number, Function Space, Wave Function.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Quantum_state Retrieved:2014-10-14.
- In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a Hilbert space, called the state vector. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vector is identified by the principal quantum number [math]\displaystyle{ \{ n \} }[/math]. For a more complicated case, consider Bohm's formulation of the EPR experiment, where the state vector :[math]\displaystyle{ \left|\psi\right\rang = \frac{1}{\sqrt{2}}\bigg(\left|\uparrow\downarrow\right\rang - \left|\downarrow\uparrow\right\rang \bigg) }[/math]
involves superposition of joint spin states for two particles. More generally, a quantum state can be either pure or mixed. The above example is pure. Mathematically, a pure quantum state is represented by a state vector in a Hilbert space over complex numbers, which is a generalization of our more usual three-dimensional space. If this Hilbert space is represented as a function space, then its elements are called wave functions.
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. Quantum states, mixed as well as pure, are described by so-called density matrices.
For example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional. A pure state here is represented by a two-dimensional complex vector [math]\displaystyle{ (\alpha, \beta) }[/math], with a length of one; that is, with :[math]\displaystyle{ |\alpha|^2 + |\beta|^2 = 1, }[/math]
where [math]\displaystyle{ |\alpha| }[/math] and [math]\displaystyle{ |\beta| }[/math] are the absolute values of [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math]. A mixed state, in this case, is a [math]\displaystyle{ 2 \times 2 }[/math] matrix that is Hermitian, positive-definite, and has trace 1.
Before a particular measurement is performed on a quantum system, the theory usually gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. These probability distributions arise for both mixed states and pure states: it is impossible in quantum mechanics (unlike classical mechanics) to prepare a state in which all properties of the system are fixed and certain. This is exemplified by the uncertainty principle, and reflects a core difference between classical and quantum physics. Even in quantum theory, however, for every observablethere are states that determine its value exactly.
- In quantum physics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a Hilbert space, called the state vector. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vector is identified by the principal quantum number [math]\displaystyle{ \{ n \} }[/math]. For a more complicated case, consider Bohm's formulation of the EPR experiment, where the state vector :[math]\displaystyle{ \left|\psi\right\rang = \frac{1}{\sqrt{2}}\bigg(\left|\uparrow\downarrow\right\rang - \left|\downarrow\uparrow\right\rang \bigg) }[/math]