Quantum mechanics Definition and 994 Threads

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.
Classical physics, the description of physics that existed before the theory of relativity and quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, while quantum mechanics explains the aspects of nature at small (atomic and subatomic) scales, for which classical mechanics is insufficient. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale.Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values (quantization), objects have characteristics of both particles and waves (wave-particle duality), and there are limits to how accurately the value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle).
Quantum mechanics arose gradually from theories to explain observations which could not be reconciled with classical physics, such as Max Planck's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein's 1905 paper which explained the photoelectric effect. These early attempts to understand microscopic phenomena, now known as the "old quantum theory", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born and others. The modern theory is formulated in various specially developed mathematical formalisms. In one of them, a mathematical entity called the wave function provides information, in the form of probability amplitudes, about what measurements of a particle's energy, momentum, and other physical properties may yield.

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  1. Danny Boy

    A Forming a unitary operator from measurement operators

    If we consider a measurement of a two level quantum system made by using a probe system followed then by a von Neumann measurement on the probe, how could we determine the unitary operator that must be applied to this system (and probe) to accomplish the given measurement operators.
  2. P

    A Localized states and density of states

    Hello, Let's suppose we have a two dimensional lattice which is periodic along certain direction, say x-direction, allowing us to define a quasi momentum k_x. The lattice is not periodic along the y-direction (perpendicular to x-direction). Therefore, we are able to obtain the band structure...
  3. J

    I Uniting Quantum Mechanics & General Relativity: 3 Forces & Geometry

    Physicists try to unite Quantum Mechanics and General Relativity. QM deals with three forces in nature (i.e., strong, weak , electromagnetism), while GR deals with geometry of space. How can one unite 3 forces and geometry? If one thinks of gravity as a force (not geometry), would one have a...
  4. R

    I Are special relativity rules encoded in the Dirac equation?

    This may seem like a stupid question, but i can't get my head around this so please bear with me. I just looked at the derivation of Dirac equation and my question is: do the solutions for a free particle obey special relativity? because if yes why? I mean I thought using E2=(mc2)2+(pc)2 would...
  5. bhobba

    A Rigged Hilbert Spaces In Quantum Mechanics

    In discussing stuff in another thread I used the standard Dirac notion expanding a state in position eigenvectors namely |u> = ∫f(x) |x>. By definition f(x) is the wave-function. I omitted the dx which is my bad but the following question was posed which I think deserved a complete answer. It...
  6. E

    I Quantum mechanics and gravity of Greenberger

    I read paper https://arxiv.org/ftp/arxiv/papers/1011/1011.3719.pdf . I do not understand figure 2. Such double phase space (x-p) can be also for a harmonic oscillator. But, at a harmonic oscillator we cannot have two ellipses (or a circle and an ellipse) which touch on some points, but have the...
  7. G

    A What does a CPT transformation do to particle properties?

    If I have a particle with: Momentum: p Spin: s Energy: E Position: x Time coordinate: t Charge: q And I preform a CPT transformation on said particle, what will these variables become? Can you show me mathematically? Also, could you show me how this effects the wavefunction/quantum state of...
  8. hilbert2

    Insights Damped Motion in Classical and Quantum Mechanics - Comments

    hilbert2 submitted a new PF Insights post Damped Motion in Classical and Quantum Mechanics Continue reading the Original PF Insights Post.
  9. MSparks6

    B How can something be "Zero Dimensional?"

    I am not a physicist by trade nor do I have any experience other than what I've read over the past few years. I once read a book by Stephen Hawking or Michio Kaku (or maybe Brian Green, I can't remember) in which they said something to the effect of "even if we had the technology to see down to...
  10. SherLOCKed

    A Operation of Hamiltonian roots on wave functions

    How come a+a- ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.
  11. SherLOCKed

    A Help with proof of eq. 2.64 of Intro. to Quantum Mechanics

    I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47. While the book has given a proof for eq. 2.64 but its not very ellaborate Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody...
  12. Jason C

    Can Zeta ζ(½+it) be interpreted as a Wave function?

    In a recent article by BBM in Physical Review Letters highlights another approach to link QM to Zeta to Prove R.H. There approach proved unsuccessful. I want to ask professional Physicists if the following new approach have merit in connecting the Zeta function to QM? This new line of attack...
  13. S

    A Operators in Quantum Mechanics

    Hey guys, Am facing an issue, we know that x and y operators take the same form due to isotropy of space, but sir if we destroy the isotropy, then what form will it take? Can u pleases throw some light on this! Thanks in advance
  14. E

    I General Questions Related to Quantum Measurement

    I've recently been wondering about the notion of measurement in quantum mechanics. While I'm aware that understanding and precisely defining the concept of a quantum mechanical measurement is at the heart of interpreting the subject, I would like some feedback on some aspects of this topic I...
  15. S

    Quantum Mechanics; Expectation value

    Homework Statement At t=0, the system is in the state . What is the expectation value of the energy at t=0? I'm not sure if this is straight forward scalar multiplication, surprised if it was, but we didn't cover this in class really, just glossed through it. If someone could walk me through...
  16. Dusty912

    Anyone here study quantum mechanics for fun?

    Just curious if any of you out there who have completely different fields study this topic just for the joy of it. It seems like you would have to be pretty motivated or extremely intelligent to master these concepts for leisure.
  17. G

    Invariance under SU(2) in quantum mechanics

    Homework Statement Hi, I'm trying to self-study quantum mechanics, with a special interest for the group-theoretical aspect of it. I found in the internet some lecture notes from Professor Woit that I fouund interesting, so I decided to use them as my guide. Unfortunately I'm now stuck at a...
  18. S

    How can Many Worlds be reconciled with unequal probabilities

    If I understand the many worlds QM interpretation correctly, for every quantum event the universe divides. However not all quantum events have a 50/50 probability. How does the many worlds interpretation deal with quantum events that have,say a 1/3 2/3 probability split?
  19. Z

    Calculate reaction heat from quantum mechanics

    Homework Statement I have some doubts about the method constructing chemical reaction process of quantum mechanics in the referencehttp://www.southampton.ac.uk/assets/centresresearch/documents/compchem/DFT_L2.pdf, for the example of ##H_2O## molecular dissociation to ##H^+## and ##OH^-## ions...
  20. R

    Quantum Mechanics - Question about the Finite Square Well

    Hi, I'm preparing for an exam, and I'm going over past papers. I've solved parts a & b of this question without any problems, however I'm finding it hard to understand part c. I thought of shifting the boundary conditions so I'd have 0 and L in the place of ± L/2, but that would not work...
  21. L

    I How to understand the derivation for this process in QFT?

    I'm reading the book "Quantum Field Theory and the Standard Model" by Matthew Schwartz and I'm finding it quite hard to understand one derivation he does. It is actually short - two pages - so I find it instructive to post the pages here: The point is that the author is doing this derivation...
  22. S

    Particle Transition between Quantum Mechanics and QFT

    Hello! I just started reading a book about QFT by Peskin (it was recommended by one of my physics professor and I saw that MIT course on QFT also uses it). However they start right away with Klein-Gordon equation suggesting that I should be familiar with it. I took 2 classes on quantum mechanics...
  23. M

    What Are the Conditions for Treating Atomic Motion Classically?

    Hello, I need same help with the following exercise: (1a)Recall Ehrenfest’s theorem and state the conditions for classicality of the trajectory of a quantum particle. (1b) Consider an atom whose state is described by a wavepacket with variance ∆x^2 in position and ∆p^2 in momentum. The atom...
  24. S

    I Is there a limit to the possible masses of particles in the universe?

    Hello! I read that, according to quantum mechanics, any physical process must happen with a certain probability if it is not forbidden by symmetry/conservation laws. And now I just started an internship at the CMS aiming to find a Higgs like particle with mass higher that 125 GeV. And we have...
  25. A

    B Is Quantum Mechanics Necessary for Understanding Water's Properties?

    To what extent is quantum mechanics needed to explain the properties of water? Thanks in advance.
  26. S

    Difference between probability waves & electromagnetic waves?

    What I know: A ripple/wave in a field gives rise to a particle. For example, a ripple in electric field creates a photon. Question: Is this the same principle as probability wave which when observed reveals a particle?
  27. A

    Quantum mechanics - Heisenberg picture

    Homework Statement We consider the O2- molecule, with the Hamiltonian and position operator having matrix representations in terms of the Pauli matrices: In the Heisenberg picture, the position operator is: (1) Find the eigenvalues and eigenstates of x(t) at time t=pi*hbar/(4A) (2) The...
  28. D

    I They physics of phase inversion in Grover's algorithm

    How would this operator be implemented physically if we had a quantum computer? In Grover's algorithm this magical operator is often called "phase inversion". Here is the operator from wiki: https://wikimedia.org/api/rest_v1/media/math/render/svg/07fb23bffa787430b084971c6a108a8f6ff6c2b3 It’s...
  29. A

    I Momentum and energy in QM and QFT

    Hi all - apologies, I'm starting a new thread here for something buried at the end of another thread - but I think the topic of that thread had changed sufficiently to warrant a more succinct top-level post. Thanks very much to PeterDonis for his very useful answers in the previous thread...
  30. MichPod

    I What's wrong with Quantum Mechanics and its teaching?

    I am afraid I have no enough competence to raise this topic. I should not have. Because of a very limited experience I cannot back up my arguments. Because of knowing mostly my own experience I should have not made generalizations. Because of not reading completely even one textbook of QM, I...
  31. S

    Shankar Exercise 1.1.3 (Vector Spaces)

    <Mentor's note: moved from a technical forum, therefore no template.> I'm long out of college and trying to teach myself QM out of Shankar's. I'm trying to understand the reasoning here because I think that I am missing something... 1.1.3 1) Do functions that vanish at the endpoints x=0 and...
  32. ShayanJ

    Insights Entanglement Entropy - Part 1: Quantum Mechanics - Comments

    ShayanJ submitted a new PF Insights post Entanglement Entropy - Part 1: Quantum Mechanics Continue reading the Original PF Insights Post.
  33. C

    What happens mathematically when a wave function collapses

    In Shankars "Principle of Quantum Mechanics" in Chapter 4, page 122, he explains what the "Collapse of the State Vector" means. I get that upon measurement, the wave function can be written as a linear combination of the eigenvectors belonging to a operator which corresponds to the...
  34. K

    A Total derivative of momentum in quantum mechanics

    In quantum mechanics, the velocity field which governs phase space, takes the form \begin{equation} \boldsymbol{\mathcal{w}}=\begin{pmatrix}\partial_tx\\\partial_tp\end{pmatrix} =\frac{1}{W}\begin{pmatrix}J_x\\J_p\end{pmatrix}...
  35. L

    Partial traces of density operators in the tensor product

    Homework Statement Consider a system formed by particles (1) and (2) of same mass which do not interact among themselves and that are placed in a potential of infinite well type with width a. Let H(1) and H(2) be the individual hamiltonians and denote |\varphi_n(1)\rangle and...
  36. G

    Relation between the matrix elements of the density matrix

    Hi. I must prove that, in general, the following relation is valid for the elements of a density matrix \rho_{ii}\rho_{jj} \geq |\rho_{ij}|^{2}. I did it for a 2x2 matrix. The density matrix is given by \rho = \left[ \begin{array}{cc} \rho_{11} & \rho_{12} \\ \rho^{\ast}_{12} & \rho_{22}...
  37. G

    I Decomposing a density matrix of a mixed ensemble

    I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...
  38. R

    Studying Tips on learning quantum mechanics

    I'm just starting my journey into physics. I am looking for good quantum mechanics books or articles if anyone has some ideas be sure to let me know.
  39. jedishrfu

    I Quantum Mechanics does Riemann Zeta Function

    "Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system." https://www.quantamagazine.org/20170404-quantum-physicists-attack-the-riemann-hypothesis/
  40. A

    I Exploring Quantum Mechanics: A Visual Journey

    This is an interesting QM video. It would be interesting to get opinions about it from the QM students and experts alike.
  41. tom.stoer

    I Interpretation of the photoelectric effect

    The photoelectric effect is usually presented as an example disproving classical electromagnetism as viable model for interaction of light with matter and as evidence of quantization of energy in the electromagnetic field, i.e. the existence of photons. I would like to discuss a thought based on...
  42. V

    Fourier transform of periodic potential in crystal lattice

    Homework Statement Homework Equations I'm not sure. The Attempt at a Solution I started on (i) -- this is where I've gotten so far. I am asked to compute the Fourier transform of a periodic potential, ##V(x)=\beta \cos(\frac{2\pi x}{a})## such that...
  43. B

    Coherent state problem in quantum mechanics

    Homework Statement Homework EquationsThe Attempt at a Solution These are what i solved except (e), I have no idea how to solve (e).
  44. J

    Optical Tweezers in the Ray Optics Regime

    Homework Statement Consider an optical tweezer operated in the ray optics regime as shown in Figure. Net reflection angle of the ray shown in Figure is θ = 10◦ (See Figure). Since more photons take the thick ray than the thin ray, a net force of F = 1 pN is exerted on the sphere toward the...
  45. Tbonewillsone

    Deriving an influence functional is deriving me mad (Help)

    Homework Statement I am attempting to derive Caldeira-Leggett's influence functional found in their paper "Path Integral Approach To Quantum Brownian Motion". If you find my following statements confusing, then pages 16-18 of http://web.science.uu.nl/itf/Teaching/2006/MxWakker.pdf show the...
  46. J

    B Quantum mechanics and the macroscopic

    I know the key tenants of quantum mechanics, and am not interested in the measurement problem. What we do know is that a particle has a wave function that describes the likelihoods of it having certain EXACT values when we measure it. That's all good, but I am still confused how the classical...
  47. N

    I Classical v. quantum dynamics: Is spin the key difference?

    I'm interested in understanding the key physical differences between classical and quantum dynamics. I understand that spin (intrinsic angular momentum) is one major physical difference.* So I wonder whether all else flows from this? Or are there other major (unrelated) physical differences...
  48. A

    Quantum I'm looking for a Quantum Mechanics textbook

    The title kind of says it all, but I'm searching for a textbook that has a heavy theoretical approach. Could someone help me with a recommendation?
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