In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle’s energy,
momentum, and other physical properties may yield.
 Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization),
objects have characteristics of both particles and waves (wave–particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions
(the uncertainty principle).
One method, called perturbation theory, uses the analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for
example) the addition of a weak potential energy.
 It is not possible to present these concepts in more than a superficial way without introducing the actual mathematics involved; understanding quantum mechanics requires
not only manipulating complex numbers, but also linear algebra, differential equations, group theory, and other more advanced subjects.
The constant is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the
ability to make such an approximation in certain limits is called the correspondence principle.
 Applications Quantum mechanics has had enormous success in explaining many of the features of our universe, with regards to small-scale and discrete quantities and interactions
which cannot be explained by classical methods.
 When quantum systems interact, the result can be the creation of quantum entanglement: their properties become so intertwined that a description of the whole solely in
terms of the individual parts is no longer possible.
Another method is called “semi-classical equation of motion”, which applies to systems for which quantum mechanics produces only small deviations from classical behavior.
The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible
to have a precise prediction for a measurement of its position and also at the same time for a measurement of its momentum.
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.
: 77–78 For the one-dimensional case in the direction, the time-independent Schrödinger equation may be written With the differential operator defined by the previous
equation is evocative of the classic kinetic energy analogue, with state in this case having energy coincident with the kinetic energy of the particle.
A simpler approach, one that has been used since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical
 Another possibility opened by entanglement is testing for “hidden variables”, hypothetical properties more fundamental than the quantities addressed in quantum theory
itself, knowledge of which would allow more exact predictions than quantum theory can provide.
A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field (rather than a fixed set of particles).
An important guide for making these choices is the correspondence principle, a heuristic which states that the predictions of quantum mechanics reduce to those of classical
mechanics in the regime of large quantum numbers.
The simplest example of quantum system with a position degree of freedom is a free particle in a single spatial dimension.
Another counter-intuitive phenomenon predicted by quantum mechanics is quantum tunnelling: a particle that goes up against a potential barrier can cross it, even if its kinetic
energy is smaller than the maximum of the potential.
A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation
is given by which is a superposition of all possible plane waves , which are eigenstates of the momentum operator with momentum .
As in the classical case, the potential for the quantum harmonic oscillator is given by This problem can either be treated by directly solving the Schrödinger equation, which
is not trivial, or by using the more elegant “ladder method” first proposed by Paul Dirac.
[note 5] Many macroscopic properties of a classical system are a direct consequence of the quantum behavior of its parts.
The commutator of these two operators is and this provides the lower bound on the product of standard deviations: 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.
For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space a probability amplitude.
Newer interpretations of quantum mechanics have been formulated that do away with the concept of “wave function collapse” (see, for example, the many-worlds interpretation).
 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.
A collection of results, most significantly Bell’s theorem, have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics.
This time evolution is deterministic in the sense that – given an initial quantum state – it makes a definite prediction of what the quantum state will be at any later time.
The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that the original quantum system ceases
to exist as an independent entity.
The angular momentum and energy are quantized and take only discrete values like those shown (as is the case for resonant frequencies in acoustics) Some wave functions produce
probability distributions that are independent of time, such as eigenstates of the Hamiltonian.
Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference, which is often illustrated with the double-slit experiment.
 The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems.
This vector is postulated to be normalized under the Hilbert space inner product, that is, it obeys , and it is well-defined up to a complex number of modulus 1 (the global
phase), that is, and represent the same physical system.
For example, the stability of bulk matter (consisting of atoms and molecules which would quickly collapse under electric forces alone), the rigidity of solids, and the mechanical,
thermal, chemical, optical and magnetic properties of matter are all results of the interaction of electric charges under the rules of quantum mechanics.
Examples Free particle Main article: Free particle Position space probability density of a Gaussian wave packet moving in one dimension in free space.
According to Bell’s theorem, if nature actually operates in accord with any theory of local hidden variables, then the results of a Bell test will be constrained in a particular,
One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone.
Instead, the wave function must satisfy more complicated mathematical boundary conditions as it is nonzero in regions outside the well.
The finite potential well problem is mathematically more complicated than the infinite particle-in-a-box problem as the wave function is not pinned to zero at the walls of
Quantum decoherence is a mechanism through which quantum systems lose coherence, and thus become incapable of displaying many typically quantum effects: quantum superpositions
become simply probabilistic mixtures, and quantum entanglement becomes simply classical correlations.
Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to
This is one of the most difficult aspects of quantum systems to understand.
Complications arise with chaotic systems, which do not have good quantum numbers, and quantum chaos studies the relationship between classical and quantum descriptions in
 One can also start from an established classical model of a particular system, and then try to guess the underlying quantum model that would give rise to the classical
model in the correspondence limit.
 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.
The Hilbert space of the composite system is then If the state for the first system is the vector and the state for the second system is , then the state of the composite
system is Not all states in the joint Hilbert space can be written in this form, however, because the superposition principle implies that linear combinations of these “separable” or “product states” are also valid.
 Particle in a box 1-dimensional potential energy box (or infinite potential well) Main article: Particle in a box The particle in a one-dimensional potential energy box
is the most mathematically simple example where restraints lead to the quantization of energy levels.
Quantum electrodynamics is, along with general relativity, one of the most accurate physical theories ever devised.
This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic.
This “semi-classical” approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles.
Therefore, since , must be an integer multiple of , This constraint on implies a constraint on the energy levels, yielding A finite potential well is the generalization of
the infinite potential well problem to potential wells having finite depth.
Another related problem is that of the rectangular potential barrier, which furnishes a model for the quantum tunneling effect that plays an important role in the performance
of modern technologies such as flash memory and scanning tunneling microscopy.
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 , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared
term is replaced with a Laplacian times .
Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as Max Planck’s solution in 1900 to the
black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein’s 1905 paper which explained the photoelectric effect.
It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner’s friend,
and its application to the universe as a whole remains speculative.
The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable
functions , while the Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors with the usual inner product.
These can be chosen appropriately in order to obtain a quantitative description of a quantum system, a necessary step in making physical predictions.
 However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves; the interference pattern appears via the
varying density of these particle hits on the screen.
This is the best the theory can do; it cannot say for certain where the electron will be found.
Defining the uncertainty for an observable by a standard deviation, we have and likewise for the momentum: The uncertainty principle states that Either standard deviation
can in principle be made arbitrarily small, but not both simultaneously.
Relation to other scientific theories Classical mechanics The rules of quantum mechanics assert that the state space of a system is a Hilbert space and that observables of
the system are Hermitian operators acting on vectors in that space – although they do not tell us which Hilbert space or which operators.
In the continuous case, these formulas give instead the probability density.
Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but
is not sufficient for describing them at small (atomic and subatomic) scales.
This approach is particularly important in the field of quantum chaos.
As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it).
Many systems that are treated dynamically in classical mechanics are described by such “static” wave functions.
The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction.
 As described above, entanglement is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured.
Such wave functions are directly comparable to Chladni’s figures of acoustic modes of vibration in classical physics and are modes of oscillation as well, possessing a sharp
energy and thus, a definite frequency.
[‘1. See, for example, Precision tests of QED. The relativistic refinement of quantum mechanics known as quantum electrodynamics (QED) has been shown to agree with experiment to within 1 part in 108 for some atomic properties.
2. ^ Physicist John
C. Baez cautions, “there’s no way to understand the interpretation of quantum mechanics without also being able to solve quantum mechanics problems – to understand the theory, you need to be able to use it (and vice versa)”. Carl Sagan outlined
the “mathematical underpinning” of quantum mechanics and wrote, “For most physics students, this might occupy them from, say, third grade to early graduate school – roughly 15 years. […] The job of the popularizer of science, trying to get across
some idea of quantum mechanics to a general audience that has not gone through these initiation rites, is daunting. Indeed, there are no successful popularizations of quantum mechanics in my opinion – partly for this reason.”
3. ^ A momentum
eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise, a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all. Consequently,
neither can belong to the particle’s Hilbert space. Physicists sometimes introduce fictitious “bases” for a Hilbert space comprising elements outside that space. These are invented for calculational convenience and do not represent physical states.: 100–105
See, for example, the Feynman Lectures on Physics for some of the technological applications which use quantum mechanics, e.g., transistors (vol III, pp. 14–11 ff), integrated circuits, which are follow-on technology in solid-state physics (vol II,
pp. 8–6), and lasers (vol III, pp. 9–13).
5. ^ see macroscopic quantum phenomena, Bose–Einstein condensate, and Quantum machine
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