-as of [20 SEPTEMBER 2024]–
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-in ‘quantum physics’, quantum state refers to the state of an isolated ‘quantum system’-
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(a ‘quantum state’ provides a ‘probability distribution’ for the ‘value’ of each ‘observable’ (i.e. for the ‘outcome’ of each ‘possible measurement’ on the ‘system’))
(knowledge of the ‘quantum state’ together with the rules for the system’s evolution in ‘time’ exhausts all that can be predicted about the system’s behavior)
(a ‘mixture’ of ‘quantum states’ is again a ‘quantum state’)
(‘quantum states’ that cannot be ‘written’ as a ‘mixture’ of other ‘states’ are called pure quantum states, all other states are called “mixed quantum states”)
(mathematically, a pure quantum state can be represented by a ‘ray’ in a ‘hilbert space’ over the ‘complex numbers’)
(the ‘ray’ is a ‘set’ of ‘nonzero vectors’ differing by just a ‘complex scalar factor’; any of them can be chosen as a state vector to represent the ‘ray’ and thus the ‘state’)
(a ‘unit vector’ is usually ‘picked’, but its ‘phase factor’ can be chosen freely anyway)
(nevertheless, such ‘factors’ are important when ‘state vectors’ are added together to form a ‘superposition’)
(‘hilbert space’ is a generalization of the ordinary ‘euclidean space’ and it contains “all possible ‘pure quantum states’ of the ‘given system'”)
(if this ‘hilbert space’, by choice of ‘representation’ (essentially a choice of ‘basis’ corresponding to a ‘complete set of observables’), is exhibited as a ‘function space’ (a ‘hilbert space’ in its own right), then the representatives are called ‘wave functions’)
(for example, when dealing with the ‘energy spectrum’ of the ‘electron’ in a ‘hydrogen atom’, the relevant ‘state vectors’ are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component sz)
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(a more complicated case is given (in ‘bra–ket notation’) by the ‘spin part’ of a ‘state vector’…)
- |ψ⟩=12(|↑↓⟩−|↓↑⟩),
which involves ‘superposition’ of ‘joint spin states’ for 2 particles with spin 1⁄2)
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(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)
(‘mixed states’ are described by so-called ‘density matrices’)
(a ‘pure state’ can also be recast as a ‘density matrix’; in this way, ‘pure states’ can be represented as a ‘subset’ of the more general ‘mixed states’)
(for example, if the ‘spin’ of an ‘electron’ is measured in any direction, e.g. with a ‘stern–gerlach experiment’, there are 2 possible results: ‘up’ or ‘down’)
(the ‘hilbert space’ for the electron’s spin is therefore 2-dimensional)
(a ‘pure state’ here is represented by a ‘2-dimensional complex vector’ (α,β), with a length of 1; that is, with
- |α|2+|β|2=1,
(where |α| and |β| are the absolute values of α and β)
(a ‘mixed state’, in this case, is a 2×2 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 ‘observable’ there are some ‘states’ that have an ‘exact’ and ‘determined’ value for that ‘observable’)
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*👨🔬🕵️♀️🙇♀️*SKETCHES*🙇♂️👩🔬🕵️♂️*
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*🌈✨ *TABLE OF CONTENTS* ✨🌷*
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🔥🔥🔥🔥🔥🔥*we won the war* 🔥🔥🔥🔥🔥🔥