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The uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position and momentum , can be simultaneously known.
Heisenberg’s uncertainty principle is a key principle in quantum mechanics. Very roughly, it states that if we know everything about where a particle is located (the uncertainty of position is small), we know nothing about its momentum (the uncertainty of momentum is large), and vice versa.
- [X; P ] = i ̄h
- < X > X ̄ = (Ψ; XΨ):
- ∆O √(Ψ; (O2 O2)Ψ); ̄
- j j jj jj jjjj
- X′ = X XI; P ′ = P P ̄ I: ̄
- ∆X ∆P ̄h 2
- S S
- P P
- P S
- S P
- Xs(0):
- Xs(0))2 >
(1) While we need this operator equation to derive a concrete result, the general idea is present in any system where there are plane waves. A physically realizable wave is always in the form of a wave packet which is finite in extent. A wave packet is built up by superposing waves with definite wave number. By simple Fourier analysis, a highly loc...
Recall that the length of a vector in Hilbert space is defined by Ψ √(Ψ; Ψ) jj jj We will also need the uncertainty in X and P: For any observable O; the standard definition of uncertainty is
which is a measure of the “spread” of values of O around its mean or expected value. To make progress in the derivation, we need an expression which can be manipulated into one involving [X; P ]: The expression used by Von Neumann is (XΨ; P Ψ): Using the fact that X and P are self-adjoint, can write 2iIm[(XΨ; P Ψ)] = (XΨ; P Ψ)
which is the Hilbert space version of the familiar statement that the dot product of two vectors is the product of their lengths. For us, the vectors we will apply the Schwarz inequality to are XΨ; and P Ψ: We have
These operators satisfy the basic commutation rule,
This completes the mathematical derivation of the uncertainty principle. There is a very nice physical discussion of it in Heisenberg’s book The Physical Principles of Quantum Theory around page 20. As a historical note, Heisenberg only proved his principle for gaussian wave functions. The full Hilbert Space derivation given above was apparently fi...
momentum Ps; which satisfy [Xs; Pp] = i ̄h: The measurement of say Xs is done by coupling the quantum system to another system, S the probe : In real experiments, will have many degrees of freedom. The final result
of the interaction of with will be a record or reading. This final stage is classical,
no issue of non-commuting operators or quantum fluctuations is involved at the end, i.e. the meter reads a certain number, period. Theoretical analysis of measurement uncertainty involving a system and a probe
have so far not included the case where has many degrees of freedom, the last stages P of which are classical. Instead the probe is modeled as a second quantum system, whose quantum variables are somehow accessible to observation, and which can give information about the quantum variables of : This can be illustrated by denoting the quantum
As a measure of the noise, the uncertainty or standard deviation of N is used,
In general, the momentum Ps of the system will be modified or disturbed by the inter-action of with : The disturbance operator is defined by P = Ps(T )
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Aug 16, 2024 · Uncertainty principle, statement that the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. The very concepts of exact position and exact velocity together have no meaning in nature. Werner Heisenberg first stated the principle in 1927.
- The Editors of Encyclopaedia Britannica
Oct 8, 2001 · The uncertainty principle is certainly one of the most famous and important aspects of quantum mechanics. It has often been regarded as the most distinctive feature in which quantum mechanics differs from classical theories of the physical world.
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Apr 14, 2021 · Key points. Applying the uncertainty principle to relationships reveals that focusing on a partner's behavior obscures objective evaluation of our own. Theory of mind enables us to predict...
