Article Outline
Glossary
Definition of the Subject
Introduction
Quantum Postulates and Associated Propositions
Dirac's Bra and Ket Notations and the Realization of Wave Function
Realization of q-Representation in Quantum Mechanics
Fourier Transformation and p‑Representation
Quantum Uncertainty and Non-Compatibility of Observables P and X
Conclusion
Future Directions
Bibliography
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Abbreviations
- Newtonian mechanics:
-
The classical mechanics based on Newton's law of motion.
- Uncertainty :
-
Also commonly referred to as error; deviation from the average value.
- Extrinsic uncertainty:
-
The uncertainty due to the systematic error and random error.
- Intrinsic uncertainty:
-
The uncertainty due to the nature of particle‐wave duality in the quantum system.
- \( { \psi (\boldsymbol{r}) } \) :
-
q‑representation of the quantum state, called state function or wave function in coordinate space.
- \( { \varphi (\boldsymbol{p}) } \) :
-
p‑representation of the quantum state, called the state function or the wave function in momentum space.
- \( { \rho (\boldsymbol{r}) } \) :
-
Probability density in coordinate space, defined as \( { \rho (\boldsymbol{r})=\psi ^{\ast} (\boldsymbol{r})\psi (\boldsymbol{r})=\left| {\psi (\boldsymbol{r})} \right|^2 } \).
- \( { \tilde {\rho} (\boldsymbol{p}) } \) :
-
Probability density in momentum space, defined as \( { \tilde {\rho} (\boldsymbol{p})=\varphi ^\ast (\boldsymbol{p})\varphi (\boldsymbol{p})=\left| {\varphi (\boldsymbol{p})} \right|^2 } \).
- \( { \vert {\psi (\boldsymbol{r})} \vert} \) :
-
Probability amplitude of state \( { \psi (\boldsymbol{r}) } \).
- Particle-wave duality:
-
A quantum object exhibits both the nature of a particle and a wave.
- Wave vector:
-
\( { \boldsymbol{k}=\boldsymbol{p}/\hbar } \).
- Wave number:
-
One‐dimensional wave vector, \( { k=p/\hbar } \).
- Quantum state:
-
The physical state in a subatomic quantum system.
- 0th Postulate of quantum mechanics:
-
Regarding quantum states as elements of Hilbert space \( { \mathcal{H} } \).
- 1st Postulate of quantum mechanics:
-
Assigning each dynamical variable in a quantum system a unique linear Hermitean operator in Hilbert space \( { \mathcal{H} } \).
- 2nd Postulate of quantum mechanics:
-
The set of eigenvectors of a given observable forms the bases of a Hilbert space.
- 3rd Postulate of quantum mechanics:
-
Poisson brackets in classical mechanics are replaced by commutators of the corresponding observables according to the relations in Eq. (5)
- [\( R, S \)]:
-
Commutator of operator R and operator S defined as \( { RS-SR } \).
- \( { \{R, S\} } \) :
-
Anti‐commutator of operator R and operator S defined as \( { RS+SR } \).
- Eigenvalue and eigenvector:
-
For operator A acting upon a particular state \( { \psi _a } \), such that \( { A \psi _a = a\psi _a} \), then the scalar number a and the state \( { \psi _a} \) are called respectively the eigenvalue and eigenvector of operator A.
- Inner product:
-
A numerically valued function of the ordered pair of vectors ψ and φ, denoted by \( { (\psi ,\varphi ) } \) such that \( { (\psi ,\varphi )=(\varphi ,\psi )^\ast} \). In Dirac's notation, it takes the forms \( { \langle \psi \vert \varphi \rangle =\langle \varphi \vert \psi \rangle ^\ast} \).
- Hilbert space:
-
A complete vector space with norm defined as the inner product.
- Dual space:
-
The space formed by the set of all functionals satisfying the linearity conditions.
- Hermitean operator:
-
An operator which is self‐adjoint; that is, an operator A equals its Adjoint conjugate, \( { A=A^{+} } \).
- Adjoint conjugate operator:
-
For an operator A and a pair of vectors ψ and φ, the Adjoint operator to A, denoted by \( { A^{+} } \) if it satisfies the relation
$$(\psi ,A\varphi )=(\varphi ,A^{+}\psi )^{\ast} =(A^{+}\psi ,\varphi )\:.$$ - \( { \vert \psi \rangle} \) :
-
A ket vector in Hilbert space.
- \( { \langle \psi \vert} \) :
-
A bra vector in dual space.
- Compatible observables:
-
Physical observables that commute to each other.
- Projection operator:
-
\( { \vert a_i \rangle \langle a_i \vert} \) a Hermitean operator that projects any vector in \( { \mathcal{H} } \) onto a subspace.
- Closure relation:
-
Direct sum of all project operators equals identity operator.
- δ‑function:
-
A distribution function denoted by \( { \delta (x-a) } \) such that
$$\begin{aligned}\int_{D} f(x)\delta (x-a)\text{d} x&=f(a)\quad\text{and}\\ \int_{D} f(x)\delta^{\prime}(x-a)\text{d} x&=-f^{\prime}(a) \quad\text{if}\enskip a\in D\:,\\\int_{D} f(x)\delta (x-a)\text{d} x&=0\quad\text{if}\enskip a\notin D\:.\end{aligned}$$ - Cauchy–Schwarz inequality:
-
Given vectors \( { \vert \alpha \rangle} \) and \( { \vert \beta \rangle} \) such that
$$\langle\alpha \vert \alpha \rangle \langle\beta \vert \beta \rangle\geq \langle \alpha \vert \beta \rangle \langle \beta \vert \alpha \rangle \:.$$ - \( { G_{\text{d}}} \) :
-
Deviation operator with respect to observable G defined as \( { G_{\text{d}}=G-\langle G\rangle I } \)
Bibliography
de Broglie L (1923) Nature 112:540. (1924) Thesis, Paris. (1925) Ann. Phys 3:22
Merzbacher E (1970) Quantum Mechanics, 2nd edn. Wiley, New York
Jordan TF (1982) Linear operators for quantum mechanics, 2nd edn. Wiley, New York
Dirac PAM (1958) The principles of quantum mechanics, 4th edn. Clarendon Press, Oxford
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Chang, KL. (2012). Granular Computing and Modeling of the Uncertainty in Quantum Mechanics . In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_87
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