Quantum mechanics provides a mathematical description of the wave-particle duality of matter and energy at small atomic and subatomic scales. It differs significantly from classical mechanics, as phenomena such as superconductivity cannot be explained using classical mechanics alone. Key aspects of quantum mechanics include wave-particle duality, the uncertainty principle, and discrete energy levels determined by Planck's constant and frequency.

1. Quantum mechanics Quantum mechanics is a branch of physics providing a mathematical description of the wavea particle duality of matter and energy. Quantum mechanics are also known as quantum physics or quantum theory. Quantum mechanics differs significantly from classical mechanics in its predictions when the scale of observations becomes comparable to the atomic and sub-atomic scale.

2. Quantum mechanics The atomic and sub-atomic scale is the so-called quantum realm. Phenomena such as superconductivity cannot be explained using classical mechanics. The wave ``particle duality'' of energy and matter and the uncertainty principle provide a unified view of the behavior of photons, electrons and other atomic-scale objects in the context of quantum mechanics. Many of the results of quantum mechanics do not have models that are easily visualized in terms of classical mechanics ; for instance , the ground state in the quantum mechanical model is a non-zero energy state that is the lowest permitted energy state of a system , rather than a traditional classical system that is thought of as simply being at rest with zero kinetic energy .

3. History Each energy element E is proportional to its frequency I: E = h nu where h is Planck's constant. Albert Einstein interpreted Planck's quantum hypothesis realistically. Albert Einstein used it to explain the photoelectric effect.

4. History Developments in quantum mechanics led to its becoming the standard formulation for atomic physics in the mid-1920s. The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta , Albert Einstein further developed this idea to show that an electromagnetic wave such as light could be described as a particle - later called the photon - with a discrete quanta of energy that was dependent on its frequency .

5. Mathematical formulations Everything appears to have a definite position, a definite momentum, a definite energy, and a definite time of occurrence. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. It is impossible to predict with certainty the result.

6. Mathematical formulations One measures the position of the particle. Observables can be either continuous or discrete. An alternative formulation of quantum mechanics is Feynman's path integral formulation. This is the quantum-mechanical counterpart of action principles in classical mechanics.

7. Interactions with other scientific theories An important guide for making these choices is the correspondence principle , which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or , equivalently , larger quantum numbers -LRB- i.e. whereas a single particle exhibits a degree of randomness , in systems incorporating millions of particles averaging takes over and , at the high energy limit , the statistical probability of random behavior approaches zero -RRB- . Classical mechanics is simply a quantum mechanics of large systems in other words. Early attempts to merge quantum mechanics with special relativity involved the replacement of the SchrAdinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation.

8. Interactions with other scientific theories This ` semi-classical ' approach fails if quantum fluctuations in the electromagnetic field play an important role by charged particles. The important role is such as in the emission of photons.

9. Quantum mechanics and classical physics Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy. \* While the seemingly exotic behavior of matter posited by quantum mechanics and relativity theory become more apparent when dealing with extremely fast-moving or extremely tiny particles , the laws of classical Newtonian physics remain accurate in predicting the behavior of the vast majority of large objectsa `` of the order of the size of large molecules and biggera `` at velocities much smaller than the velocity of light .

10. Relativity and quantum mechanics He did not accept the more philosophical consequences and interpretations that a single subatomic particle can occupy numerous areas of space at one time. He contributed to the field. The Einstein-Podolsky-Rosen paradox shows in any case that there exist experiments by which one can measure the state of one particle and instantaneously change the state of its entangled partner.

11. Relativity and quantum mechanics This effect does not violate causality. The two particles can be an arbitrary distance apart. No transfer of information happens.

12. Philosophical implications Albert Einstein disliked this loss of determinism in measurement. Albert Einstein was himself one of the founders of quantum theory. Einstein held that there should be a local hidden variable theory underlying quantum mechanics.

13. Philosophical implications Einstein held that the present theory was incomplete. Experiments have been performed confirming the accuracy of quantum mechanics. Experiments demonstrate that the physical world cannot be described by local realistic theories.

14. Philosophical implications This is not accomplished by introducing some new axiom to quantum mechanics , but on the contrary by removing the axiom of the collapse of the wave packet : All the possible consistent states of the measured system and the measuring apparatus -LRB- including the observer -RRB- are present in a real physical -LRB- not just formally mathematical , as in other interpretations -RRB- quantum superposition . This inaccessibility can be understood as follows : Once a measurement is done , the measured system becomes entangled with both the physicist who measured it and a huge number of other particles , some of which are photons flying away towards the other end of the universe ; in order to prove that the wave function did not collapse one would have to bring all these particles back and measure them again , together with the system that was measured originally .

15. Applications of quantum mechanics The individual behavior of the subatomic particles that make up all forms of mattera electrons, protons, neutrons, photons and othersa can often only be satisfactorily described using quantum mechanics. Recent work on photosynthesis has provided evidence that quantum correlations play an essential role in this most fundamental process of the plant kingdom.

16. Free particle For example, consider a free particle. This is called an eigenstate of position a generalized position eigenstate.