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The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wavefunction collapse.[2] This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe.[3]
Uncertainty principle
One consequence of the basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum.[19][20] Both position and momentum are observables, meaning that they are represented by Hermitian operators. The position operator and momentum operator {\displaystyle {\hat {P}}} do not commute, but rather satisfy the canonical commutation relation. Given a quantum state, the Born rule lets us compute expectation values for both and, and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation, we have\sigma _{X}={\sqrt {\langle {X}^{2}\rangle -\langle {X}\rangle ^{2}}},} and likewise for the momentumsigma _{P}={\sqrt {\langle {P}^{2}\rangle -\langle {P}\rangle ^{2}}}.
The uncertainty principle states that sigma _{X}\sigma _{P}\geq {\frac {\hbar }{2}.
Either standard deviation can in principle be made arbitrarily small, but not both simultaneously.[21] This inequality generalizes to arbitrary pairs of self-adjoint moderators A and B. The commutator of these two operators is [A,B]=AB-BA,and this provides the lower bound on the product of standard deviations:
sigma _{A}\sigma _{B}\geq {\frac {1}{2}}\left|\langle [A,B]\rangle \right|. Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of the dependence in position means that the momentum operator is equivalent (up to an | factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space.
This is why in quantum equations in position space, the momentum is replaced by |\frac {\partial }{\partial x}, and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times |r ^{2}}.[19]
Composite systems and entanglement
When two different quantum systems are considered together, the Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces {\displaystyle {\mathcal {H}}_{A}} and {\mathcal {H}}_{B}}, respectively. The Hilbert space of the composite system is then {\mathcal {H}}_{AB}={\mathcal {H}}_{A}\otimes {\mathcal {H}}_{B}. If the state for the first system is the vector psi _{A}} and the state for the second system is psi _{B}}, then the state of the composite system is psi _{A}\otimes \psi _{B}. Not all states in the joint Hilbert space mathcal {H}}_{AB}} can be written in this form, however, because the superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if psi _{A}} and phi _{A}} are both possible states for system {\displaystyle A}, and likewise {\displaystyle \psi _{B} and phi _{B} are both possible states for system {\displaystyle B}, then frac {1}{\sqrt {2}}\left(\psi _{A}\otimes \psi _{B}+\phi _{A}\otimes \phi _{B}\right) is a valid joint state that is not separable. States that are not separable are called entangled.[22][23]
If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes a loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system.[22][23] Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory.[22][24]
As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence. This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.[25]
Equivalence between formulations
There are many mathematically equivalent formulations of quantum mechanics. One of the oldest and most common is the "transformation theory" proposed by Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).[26] An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics.
Symmetries and conservation laws
Main article: Noether's theorem
The Hamiltonian is known as the generator of time evolution, since it defines a unitary time-evolution operator U(t)=e^{-iHt/\hbar } for each value of t. From this relation between U(t) and }, it follows that any observable A that commutes with H} will be conserved: its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A can generate a family of unitary operators parameterized by a variable t. Under the evolution generated by A, any observable B that commutes with A will be conserved. Moreover, if B is conserved by evolution under A, then A is conserved under the evolution generated by B. This implies a quantum version of the result proven by Emmy Noether in classical (Lagrangian) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law.
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