Critical look at the time-energy uncertainty relations

K. Urbanowski

REVIEW 2 major objections 4 minor 36 references

Critical look at the time-energy uncertainty relations

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper argues that time–energy uncertainty relations cannot be universal quantum laws.

desk verdict The division-by-zero caveat is real but already known; the paper's dense-set claim is false, and the right outcome is desk reject. read the letter →

arxiv 1908.05273 v1 pith:T72XTIBE submitted 2019-08-14 quant-ph hep-phphysics.pop-ph

K. Urbanowski This is my paper · ORCID

classification quant-phhep-phphysics.pop-ph PACS 03.65.-w03.65.Ta01.55.+b

keywords time-energyuncertaintyrelationMandelstam-TamminequalityuniversalvaliditystationarystatesPaulitheoremdivisionbyzeroquantumfoundationsprinciple

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper challenges the common view that time–energy uncertainty relations have the same rigorous status as the position–momentum uncertainty relation. It argues that the standard derivation of the Mandelstam–Tamm relation divides both sides by a quantity that is zero for stationary states, making the relation undefined on a linearly dense set of states. The author concludes that the Heisenberg and Mandelstam–Tamm time–energy relations cannot be considered universally valid and should be treated as conditional estimates for non-stationary systems. If correct, this narrows what can be inferred from time–energy uncertainty in quantum mechanics and in applications such as cosmology.

What carries the argument

The load-bearing object is the set $\Sigma_H$ of Hamiltonian eigenvectors, which the paper takes to be linearly dense in the state space—meaning finite linear combinations of these vectors can approximate every state. The mechanism that carries the argument is the division step in the Mandelstam–Tamm derivation: one divides by $|d\langle A\rangle_\varphi/dt|$ to define $\tau_A$. For every eigenstate of $H$, this derivative is zero and $\Delta_\varphi E$ is also zero, so the division is undefined and the inequality reduces to $0 = 0$. Since the same happens for eigenstates of $A$, the relation fails on a dense set. A supporting mechanism is the no-self-adjoint-time-operator theorem, which keeps time from being treated as an observable like position.

What would settle it

Check the Mandelstam–Tamm inequality on the free particle, whose Hamiltonian has a purely continuous spectrum and no normalizable energy eigenvectors; if the inequality holds for every normalizable state in that system, the paper's dense-set argument does not apply, because the set of eigenvectors is empty rather than dense.

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Extended reading notes

Core claim

The central claim is a negative result: within standard quantum mechanics, the time–energy uncertainty relations of type $\Delta_\varphi t \cdot \Delta_\varphi E \geq \hbar/2$ and $\tau_A \cdot \Delta_\varphi E \geq \hbar/2$ fail to be universally valid. The proof rests on the observation that in the Mandelstam–Tamm derivation, after using $\langle [A,H]\rangle_\varphi = i\hbar\, d\langle A\rangle_\varphi/dt$, one divides by $|d\langle A\rangle_\varphi/dt|$ to define the time scale $\tau_A$. For any eigenvector of $H$ (or of $A$), this derivative is zero, so the division is undefined; at the same time $\Delta_\varphi E$ is also zero, reducing the inequality to the trivial equality $0 = 0$. Because the eigenvectors of $H$ form a linearly dense set in the state space, the relation cannot hold on a dense set and therefore cannot be called a universal principle. The paper also notes that a well-known theorem rules out a self-adjoint time operator for a Hamiltonian bounded from below, further separating time from position.

Load-bearing premise

The argument depends on the premise that the eigenvectors of the Hamiltonian form a linearly dense set in the state space, and that a relation called 'universally valid' must hold even for stationary states, where the expectation value of any observable does not change in time.

Editorial extensions

If this is right

Any use of the Mandelstam–Tamm relation outside the domain where both $\Delta_\varphi H$ and $d\langle A\rangle_\varphi/dt$ are nonzero needs an explicit justification.
Time–energy uncertainty should not be cited as a universal bound comparable to position–momentum uncertainty in foundational arguments.
Physicists using time–energy relations in cosmology or astrophysics should check that the states involved are non-stationary and that the relation is being read as a conditional estimate.
For unstable states, the paper says the relevant relations (such as lifetime–width) are exact-value relations, not uncertainty relations, so they should not be used to infer a fundamental time–energy trade-off.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the linear-denseness argument generalizes, any derived inequality obtained by dividing by a quantity that can vanish on a dense set becomes a conditional estimate rather than a universal law; this may apply to other 'uncertainty' relations in the literature.
The paper's conclusion leaves open the possibility that the Mandelstam–Tamm bound is valid on the dense set of states with $\Delta_\varphi H \neq 0$; whether that counts as 'universal' is a definitional question, and the paper's choice to include stationary states is a substantive one.
A numerical study of the relation for a free particle or other continuous-spectrum Hamiltonian could determine whether the failure on stationary states is the only obstruction to universal validity.
The Planck–Einstein exact relation $T_\varphi E_\varphi = h$ suggests that many textbook 'time–energy uncertainty' applications may be re-readable as exact relations, which would change how those results are interpreted.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here is my take. The paper correctly identifies a real defect in the standard textbook route to the Mandelstam–Tamm relation: after Eq. (26), dividing by |d⟨A⟩/dt| is not allowed when that derivative vanishes, which happens for eigenstates of H or A. That observation is sound, and the paper explains it clearly. But it is not new. It is in Gray and Vogt (2005), which the paper cites, and Mandelstam and Tamm themselves explicitly restricted the relation to states with ΔE≠0. The paper then pushes the point too far. It claims relations (27)/(29) 'cannot hold on linearly dense sets' and therefore are not universally valid. That inference is invalid. Undefinedness on a set is not falsity on that set, and eigenvectors are not a dense set of counterexamples anyway. The stress-test example nails it: for H and A as given, τ_A ΔE = ℏ/2 for all θ in (0,π/2), so (29) actually holds on a linearly dense set. And for continuous-spectrum Hamiltonians, the eigenvector set contains no normalizable states, so the counterexample class is empty. The denseness argument is the load-bearing part of the paper, and it collapses.

Credit where due: the paper is honest about the Mandelstam–Tamm caveat, the stationary-state discussion is intuitive, and the Planck–Einstein rescaling in Sec. 4 is clearly labeled as heuristic. The citation pattern is fine, though leaning on the author's own preprint [35] in the conclusion is weak but minor.

Net: the paper is a well-written restatement of a known caveat with an overblown conclusion. It will not change the literature, and the novel part is wrong. A fresh reader might find the derivation walkthrough useful, but a knowledgeable referee would reject the dense-set claim. I would not bring it to the reading group, would not cite it, and would not send it to peer review. The right outcome is desk reject, with a note pointing to Gray & Vogt and the two-level counterexample.

Referee Report

2 major / 4 minor

Summary. The manuscript critically examines the Heisenberg and Mandelstam–Tamm (MT) time–energy uncertainty relations. It first reviews the standard derivation of the Robertson inequality (5) and then considers the MT step that replaces the commutator average with ℏ|d⟨A⟩_φ/dt|/2 and divides by this derivative to define τ_A. The author observes that for eigenstates of H or of A the derivative and the relevant variance vanish, so Eq. (26) becomes 0·0 ≥ 0 and the quotient defining τ_A is undefined. From this, Section 3 concludes that relations (27) and (29) cannot hold on linearly dense sets and are therefore not universally valid. Section 4 discusses stationary states and the Planck–Einstein relation, and Section 5 repeats the non-universality conclusion, also invoking the author's previous preprint [35].

Significance. If the non-universality claim were established, the paper would be significant: it would challenge the textbook status of time–energy uncertainty as a rigorous counterpart to position–momentum uncertainty. The paper has the merit of highlighting a genuine domain issue in the textbook derivation of the MT relation, namely that τ_A is not defined when d⟨A⟩/dt = 0, and the algebraic derivation up to Eq. (26) is standard. However, the paper's decisive inference from undefinedness to non-universality is not sound, as detailed below; the result, as it stands, is a caveat about the domain of the derivation rather than a proof that the relations are not universally valid. The paper contains no fitted parameters or numerical benchmarks; its argument is analytic.

major comments (2)

[Sec. 3, Eqs. (34)–(37)] The inference from the vanishing of d⟨A⟩_φ/dt on eigenstates of H or A to the conclusion that (27)/(29) 'cannot hold on linearly dense sets' is invalid. For an eigenstate |φ_β⟩ of H, Eq. (26) gives the true statement 0 · Δ_{φ_β}E ≥ 0; what fails is only the subsequent division by |d⟨A⟩_φ/dt| = 0, which makes τ_A undefined. An undefined expression is not a counterexample to an inequality. Furthermore, states on which the relation does hold can be linearly dense: for the two-level system H = E0|0⟩⟨0| + E1|1⟩⟨1|, A = |0⟩⟨1| + |1⟩⟨0|, and |ψ(θ)⟩ = cosθ|0⟩ + i sinθ|1⟩, a direct calculation gives τ_A ΔE = ℏ/2 for every θ ∈ (0,π/2). Hence the proof establishes only a domain caveat about division by zero, not non-universality.
[Sec. 4, paragraph on Mandelstam–Tamm reservation] The paper concedes in Sec. 4 that Mandelstam and Tamm themselves restricted the derivation to states with Δ_φH ≠ 0 and to eigenvectors corresponding to the continuous spectrum. This concession means the eigenvector counterexamples of Sec. 3 fall outside the domain of the original conditional relation. Moreover, for Hamiltonians with purely continuous spectrum, such as a free particle, Σ_H contains no normalizable eigenvectors, so the counterexample class can be empty. To sustain the headline claim, the manuscript must show that the conditional form τ_A Δ_φE ≥ ℏ/2 for states with Δ_φH > 0 and d⟨A⟩_φ/dt ≠ 0 fails or is unjustified; the present argument does not address that form.

minor comments (4)

[Abstract and Introduction] There are several typographical errors, including 'conlusion' in the abstract, 'Schroödinger' in the introduction, 'Mandelstm', 'operstor', 'Hesienberg's', and 'nad'; these should be corrected.
[Sec. 4, Eqs. (38) and (41)] The statement that Eqs. (38) and (41) are 'mathematically identical' is misleading: (38) is an inequality about uncertainties Δ_φt and Δ_φE, while (41) is an exact relation between the period T_φ and energy E_φ. Please rephrase this as a scaling analogy or a dimensional observation.
[Sec. 5] The conclusion states that the analysis 'together with the conclusions presented in [35]' supports the result; since [35] is the author's own preprint, the reader needs either a summary of its argument or a published reference.
[Sec. 3, 'linearly dense' terminology] The phrase 'linearly dense set' is used without a definition. Please define it explicitly and distinguish between the set itself being dense and the property (29) holding on all elements of a dense set; the current usage is ambiguous.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor non-load-bearing self-citation; central derivation is self-contained.

other [Section 5, Conclusions (final paragraph)]
"The analysis of the discussion of relations (23) and (29) in previous Sections together with the conclusions presented in [35] show that these time–energy uncertainty relations are not well founded and can not be considered as universally valid."

Reference [35] is the author's own earlier preprint on the same question, and the sentence uses it as joint support for the central conclusion. However, the Section 3 derivation from Eqs. (24)-(37) is self-contained and does not rely on [35]; no fitted parameter, external benchmark, or imported uniqueness theorem is used. Thus this is a minor self-citation signal, not a load-bearing circular step.

full rationale

The paper's main claim is a mathematical critique of the Mandelstam-Tamm derivation. The transition from Eq. (26) to (27) is the standard division by |d<A>/dt|; the paper's objection is that this quantity is zero on eigenvectors, making tau_A undefined. This is a direct argument from Eqs. (34)-(37), not a prediction of a fitted quantity. The only self-reference is the concluding citation of [35], which is not load-bearing because the Section 3 argument stands independently. The logical weakness flagged by critics, namely that undefinedness on eigenvectors does not establish failure on a dense set, is a correctness concern rather than circularity. No input parameter is renamed as a result, and no result is imported from same-author prior work to force the conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. Its argument rests on standard quantum mechanical assumptions: Hilbert-space observables, the Heisenberg equation of motion, Pauli's theorem on the absence of a self-adjoint time operator for bounded-below Hamiltonians, and the Planck-Einstein relation. The least standard premise is that energy eigenvectors form a linearly dense set for the systems where the critique is applied, which is stated but not proved and fails for continuous spectra.

assumptions (5)

domain assumption Standard Hilbert-space formalism of quantum mechanics with self-adjoint observables and root-mean-square deviations.
Section 2 defines an observable F as a hermitian operator acting in a Hilbert space and defines Delta_phi F via Eq. (3); the paper assumes this framework throughout.
domain assumption Heisenberg equation of motion: angle [A,H] angle = i hbar d angle A angle / dt.
Eq. (25) in Section 3, used to replace the commutator average with a time derivative in deriving the Mandelstam-Tamm inequality.
domain assumption Pauli's theorem: no self-adjoint time operator T satisfies [H,T] = i hbar I if H is bounded below.
Section 3 invokes Pauli's theorem to argue time cannot be an observable, establishing a different status for time-energy versus position-momentum uncertainty.
domain assumption The set of eigenvectors of the Hamiltonian is linearly dense in the state space for the systems considered.
Section 3, stated as 'usually...linearly dense (complete) set'; used to extend the eigenvector counterexample to a claim of non-universality. Not true for continuous spectra.
domain assumption Planck-Einstein relation E = h nu and nu = 1/T for photons.
Section 4, Eqs. (39)-(41), used to argue the exact-value relation conflicts with the uncertainty interpretation.

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Cite this review

Pith. "Pith review of Critical look at the time-energy uncertainty relations." pith.science (2026). https://pith.science/paper/T72XTIBE

@misc{pith2026190805273,
  author       = {Pith},
  title        = {Pith review of: Critical look at the time-energy uncertainty relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T72XTIBE}},
  note         = {Machine review of arXiv:1908.05273}
}

read the original abstract

The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schr\"{o}dinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 31 canonical work pages

[35]

Remarks on the uncertainty relations

K. Urbanowski, Time–energy uncertainty relations: Ar e they universally valid?, arXiv: 1810.11462v3 [quant-ph]

work page Pith review arXiv
[1]

Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , Zeitschrift for Physik, 43, 172 – 98 (1927)

W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , Zeitschrift for Physik, 43, 172 – 98 (1927). English translation in: Quantum theory and measurements , Eds: J.A. Wheeler and W. H. Żurek, Princeton University Press 1983

work page 1927
[2]

Heisenberg, The physical principles of the quantum theory , Dover Publications Inc., 1930

W. Heisenberg, The physical principles of the quantum theory , Dover Publications Inc., 1930

1930
[3]

H. P. Robertson, The Uncertainty Principle, Phys. Rev., 34, 163, (1929)

1929
[4]

Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber

E. Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber. Preuss. Akad. Wiss., XIX, 296 – 303, (1930) [Proceedings of The Prus- sian Academy of Sciences. Physics–Mathematical Section, XIX, 296- 303, (1930)]; English translation: About Heisenberg Uncertainty Rela- tion, Bulg. J. Phys. 26, 193–203, (1999); arXiv: 9903100 [quant–ph]

1930
[5]

Messiah, Quantum mechanics , vol

A. Messiah, Quantum mechanics , vol. I, North Holland Publ. Co., Am- sterdam, 1962

1962
[6]

Teschl, Mathematical Methods in Quantum Mechanics (With Applica- tions to Schrödinger Operators) , Graduate Studies in Mathematics Vol

G. Teschl, Mathematical Methods in Quantum Mechanics (With Applica- tions to Schrödinger Operators) , Graduate Studies in Mathematics Vol. 99, (American Mathematical Society, Providence, Rhode Isl and, 2009)

2009
[7]

Merzbacher, Quantum Mechanics , 3rd ed., New York, NY, Wiley, 1998

E. Merzbacher, Quantum Mechanics , 3rd ed., New York, NY, Wiley, 1998

work page 1998

Show all 36 references

[8]

Nouredine Zettili, Quantum Mechanics Concepts and Applications , 2nd ed., Wiley, 2009

2009
[9]

Hall, Quantum Theory for Mathematicians , Springer (New York, Heidelberg, Dordrecht, London), 2013

Brian C. Hall, Quantum Theory for Mathematicians , Springer (New York, Heidelberg, Dordrecht, London), 2013

2013
[10]

D. J. Griﬃths, An Introduction to Quantum Mechanics , Prentice Hall Inc. New Jersey, 1995. 13

1995
[11]

Michel Le Bellac, Quantum Physics , Cambridge University Press, 2006

2006
[12]

Galindo, B

A. Galindo, B. Pascual, Quantum mechanics I , Springer (1990)

1990
[13]

Jammer, The Philosophy of Quantum Mechanics , (Wiley, New York 1974)

M. Jammer, The Philosophy of Quantum Mechanics , (Wiley, New York 1974)

1974
[14]

Mandelstam and Ig

L. Mandelstam and Ig. Tamm, The Uncertainty Relation Between En- ergy and Time in Non-relativistic Quantum Mechanics , J. Phys. USSR, 9, 249 – 254 (1945), and I.E. Tamm Selected Papers , Eds. B. M. Bolo- tovskii and V. Ya. Frenkel, (Springer, Berlin 1991)

1945
[15]

Butterﬁeld, On Time in Quantum Physics , The Blackwell Companion to the Philosophy of Time, Edited by A

J. Butterﬁeld, On Time in Quantum Physics , The Blackwell Companion to the Philosophy of Time, Edited by A. Bardon and H. Dyke, Wil ey- Blackwell (2013), pp. 220–241; arXiv: 1406.4745 [physics. hist–ph]

2013 arXiv
[16]

Bauer and P

M. Bauer and P. A. Mello, On the lifetime-width relation for a decaying state and the uncertainty principle , Proc. Nat. Acad. Sci. USA, 73, 283 – 285, (1976)

1976
[17]

E. A. Gislason and N. H. Sabelli, New form of the time-energy uncer- tainty relation , Phys. Rev., A 31 , 2078, (1985)

1985
[18]

S. Kalyana Rama, Some consequences of the generalised uncertainty principle: statistical mechanical, cosmological, and var ying speed of light, Physics Letters B, 519, 103 -– 110, (2001)

2001
[19]

Pauli, General Principles of Quantum Mechanics (Spr inger- Verlag, New York 1980), p

W. Pauli, General Principles of Quantum Mechanics (Spr inger- Verlag, New York 1980), p. 63, Footnote 2

1980
[20]

Busch, The Time–Energy Uncertainty Relation , Eds

P. Busch, The Time–Energy Uncertainty Relation , Eds. G. Muga et al , Time in Quantum Mechanics, (Lect. Notes Phys. 734, Chap. 3, 7 3 ––105, (Springer–Verlag Berlin, Heidelberg 2008))

2008
[21]

Yu. I. Vorontsov, The uncertainty relation between energy and time of measurement, Sov. Phys. Usp., 24, 150, (1981)

1981
[22]

Hilgevoord, The uncertainty principle for energy and time , Am

J. Hilgevoord, The uncertainty principle for energy and time , Am. J. Phys., 64, 1451, (1996)

1996
[23]

Hilgevoord, The uncertainty principle for energy and time

J. Hilgevoord, The uncertainty principle for energy and time. II , Am. J. Phys., 66, 396, (1998). 14

1998
[24]

Briggs, A derivation of the time-energy uncertainty relation , J

S. Briggs, A derivation of the time-energy uncertainty relation , J. Phys.: Conf. Ser. 99, 012002, (2008)

2008
[25]

J. E. Gray, and A. Vogt, Mathematical analysis of the Mandelstam— Tamm time-energy uncertainty principle , Journal of Mathematical Physics, 46, 052108 (2005)

2005
[26]

Aharonov and A

Y. Aharonov and A. D. Bohm, Time in the Quantum Theory and the Un- certainty Relation for Time and Energy , Phys. Rev., 122, 1649, (1961)

1961
[27]

Misra and E

B. Misra and E. C. G. Sudarshan, The Zeno’s paradox in quantum the- ory, J. Math. Phys. 18, 756 – 763 (1977)

1977
[28]

Beige and G

A. Beige and G. C. Hegerfeldt, Projection postulate and atomic quantum Zeno eﬀect , Phys. Rev. A, 53, 53, (1996)

1996
[29]

Urbanowski, Early–time properties of quantum evolution , Phys

K. Urbanowski, Early–time properties of quantum evolution , Phys. Rev. A, 50, 2847, (1994)

1994
[30]

Koshinoa, A

K. Koshinoa, A. Shimizuc, Quantum Zeno eﬀect by general measure- ments, Physics Reports, 412, 191 – 275, (2005)

2005
[31]

Facchi and S

P. Facchi and S. Pascazio, Quantum Zeno Phenomena: Pulsed versus Continuous Measurement, Fortschr. Phys., 49, 941 – 947 (2001)

2001
[32]

Fock and N

V. Fock and N. Krylov, On the Uncertainty Relation between Time and Energy, JETP 17, 93, (1947); (J. Phys. USSR, 11, 112, (1947))

1947
[33]

Bhattacharyya, Quantum decay and the Mandelstam–Tammenergy inequality, J

K. Bhattacharyya, Quantum decay and the Mandelstam–Tammenergy inequality, J. Phys. A: Math. Gen., 16, 2993, (1983)

1983
[34]

T. B. Boykin et al, Evolution time and energy uncertainty , Eur. J. Phys., 28, 673, (2007)

2007
[36]

Maziashvili, Cosmological implications of Károlyházy uncertainty re- lation , Physics Letters B, 652, 165 -– 168, (2007); V

M. Maziashvili, Cosmological implications of Károlyházy uncertainty re- lation , Physics Letters B, 652, 165 -– 168, (2007); V. Cooray, G. Cooray, F. Rachidi, Emergence of Friedmann Equation of Cosmology of a Flat 15 Universe from the Time–Energy Uncertainty Principle , Journa...

2007 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [35]

Remarks on the uncertainty relations

K. Urbanowski, Time–energy uncertainty relations: Ar e they universally valid?, arXiv: 1810.11462v3 [quant-ph]

work page Pith review arXiv

[2] [1]

Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , Zeitschrift for Physik, 43, 172 – 98 (1927)

W. Heisenberg, Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , Zeitschrift for Physik, 43, 172 – 98 (1927). English translation in: Quantum theory and measurements , Eds: J.A. Wheeler and W. H. Żurek, Princeton University Press 1983

work page 1927

[3] [2]

Heisenberg, The physical principles of the quantum theory , Dover Publications Inc., 1930

W. Heisenberg, The physical principles of the quantum theory , Dover Publications Inc., 1930

1930

[4] [3]

H. P. Robertson, The Uncertainty Principle, Phys. Rev., 34, 163, (1929)

1929

[5] [4]

Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber

E. Schrödinger, Zum Heisenberschen Unschäfprinzip , Sitzungsber. Preuss. Akad. Wiss., XIX, 296 – 303, (1930) [Proceedings of The Prus- sian Academy of Sciences. Physics–Mathematical Section, XIX, 296- 303, (1930)]; English translation: About Heisenberg Uncertainty Rela- tion, Bulg. J. Phys. 26, 193–203, (1999); arXiv: 9903100 [quant–ph]

1930

[6] [5]

Messiah, Quantum mechanics , vol

A. Messiah, Quantum mechanics , vol. I, North Holland Publ. Co., Am- sterdam, 1962

1962

[7] [6]

Teschl, Mathematical Methods in Quantum Mechanics (With Applica- tions to Schrödinger Operators) , Graduate Studies in Mathematics Vol

G. Teschl, Mathematical Methods in Quantum Mechanics (With Applica- tions to Schrödinger Operators) , Graduate Studies in Mathematics Vol. 99, (American Mathematical Society, Providence, Rhode Isl and, 2009)

2009

[8] [7]

Merzbacher, Quantum Mechanics , 3rd ed., New York, NY, Wiley, 1998

E. Merzbacher, Quantum Mechanics , 3rd ed., New York, NY, Wiley, 1998

work page 1998