Lecture 8 - Axioms of quantum mechanics Generalized...

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Axioms of quantum mechanics Generalized Statistical Interpretation
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Quantum states Properties of quantum mechanical systems are described in terms of quantum states. A state can be described using different mathematical objects. Coordinate dependent wave functions is one example of such object. The same state, however, can be also described by a Fourier transform of the wave function, called momentum-dependent wave function. Yet another example can be constructed by presenting a coordinate wave function as a linear combination of a discrete set of functions such as eigenstates of harmonic oscillator 0 p ( ) ( ) n n n x a x ψ ϕ = In this example the set of discrete coefficients can be used to describe the same state. The most important are states, in which one or several physical quantities have definite values. Then these values can be used to specify states. For instates with definite value of momentum can be described either using plane waves or delta functions . In order to describe a state in a form independent of any particular choice of presentation we use notation like this 0 0 ( ) exp[ ] p x ip x = 0 0 ( ) ( ) p p p p δ = - Stationary states can be designated as n E
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Superposition principle , dead alive Then the cat can also be in a superposition state a dead b alive + 1 2 1 2 1, ;2, 1, 2, E E E E = In which her mortality is undetermined. In the system of several particles superposition principle predicts existence of rather weird states. For instance consider noninteracting particles such that each one of them is in some stationary state One can also envision a state 2 1 2 1 1, ;2, 1, 2, E E E E = 2 1 1 2 1, 2, 1, ;2, E E E E + In this state particle are statistically dependent on each other. (entangled) States in which particle does not have definite values of some observable can be constructed as linear combinations of the states, in which it does. The resulting state is also an allowed state of a system according to superposition principle. For instance a cat can be in one of two definite states According to superposition principle such state also exists
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Bra-ket notation for states Using the notation for the states in the form of we can also simplify notation for integrals with wave functions. n E * * ( ) ( ) ˆ ˆ ( ) ( ) x x dx x A x dx A ϕ ψ φ ψ φ We can introduce notation to represent states on which operators are supposed to act on the left
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Observables To every physical observable (any numerical quantity which can be measured) there corresponds an operator acting on states. This operator is linear and hermitian. The only
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Lecture 8 - Axioms of quantum mechanics Generalized...

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