What is Quantum mechanics: Definition and 995 Discussions

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. Graham87

    Intro Quantum Mechanics - Dirac notations

    I am learning Dirac notations in intro to quantum mechanics. I don’t understand why the up arrow changes to down arrow inside the equation in c). My own calculation looks like this:
  2. Graham87

    Quantum mechanics - Find S_x and S_y

    I have a lecture slide that shows how to find S_x and S_y. I get all the steps except the last row. Where did 1/2 come from? I think my linear algebra needs polishing. Thanks!
  3. J

    I Black hole singularity vs. quantum mechanics

    I'm wondering about some aspects about black holes (BH) and singularities, but since all my questions have to do mostly with quantum mechanics, I placed this thread in here. OK, let's assume there IS a singularity in the middle of a BH. A) Pauli exclusion principle (PEP) says no two fermions...
  4. D

    I "No objective reality" in quantum mechanics?

    As per title and the TL;DR, I'm curious if there could be some truth in these statements of the headlines I had read recently or are they just sensationalist fluff. Personally, I find these statements very hard to believe. In fact, impossible to believe. But I'm not a QM expert, not even an...
  5. Delta2

    I Earth's orbit not perfect ellipse

    Listen to the following arguments: Earth's orbit isn't perfect ellipse because classically there is the gravitational field of moon and possibly of Mars and Venus which affect it According to general relativity isn't perfect ellipse because there is the curvature of space time which doesn't...
  6. J

    I Dimensions of quantum cell automata's state space

    In the paper C. S. Lent and P. D. Tougaw, "A device architecture for computing with quantum dots," in Proceedings of the IEEE, vol. 85, no. 4, pp. 541-557, April 1997, doi: 10.1109/5.573 about quantum dots, it is stated that the basis vectors in the space of quantum states for a single cell...
  7. A. Neumaier

    A Exploring the Limits of Quantum Mechanics: David Wallace's Manuscript (2022)

    David Wallace, The sky is blue, and other reasons quantum mechanics is not underdetermined by evidence, Manuscript (2022). arXiv:2205.00568. From the Abstract: ''I argue that there as yet no empirically successful generalization of'' [Bohmian Mechanics and dynamical-collapse theories like the...
  8. Graham87

    Quantum mechanics - finite square well

    In a) I get that T should be largest where V_0 is least wide, because when V_0 is infinitely wide the particle would be fully reflected. But I don't get how height in b) and energy levels height in c) correlates to T and R. Is it because of their k? I get the opposite answer from the correct...
  9. Graham87

    Quantum mechanics - infinite square well problem

    I have solved c), but don’t know how to solve the integral in d. It looks like an integral to get c_n (photo below), but I still can’t figure out what to make of c) in the integral of d). I also thought maybe you can rewrite c) into an initial wave function (photo below) with A,x,a but don’t...
  10. Salmone

    I Strange Hamiltonian of two particles on the surface of a sphere

    I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators...
  11. warhammer

    Requesting guidance on Commutators & Intro QM

    I have approached this question step by step as shown in the image attached. I request someone to please guide if I have approached the (incomplete) solution correctly and also guide towards the complete solution, by helping me to rectify any mistakes I may have made. I'm still unsure how to...
  12. Salmone

    I Wavefunction properties tunneling effect

    I am considering tunnel effect with a potential barrier of a certain height that is ##\neq 0## only for ##0 \le x \le a## . I write the Hamiltonian eigenfunctions outside the barrier as:## \psi_E(x)=\begin{cases} e^{ikx}+Ae^{-ikx} \quad \quad x \le0 \\ Ce^{ikx} \quad \quad x\ge a \\...
  13. Salmone

    I Linear combination of states with Pauli's principle

    If I have two identical particles of ##1/2## spin, for Pauli's exclusion principle all physical states must be antysimmetrical under the exchange of the two particles, so ##\hat{\Pi}|\alpha\rangle=-|\alpha\rangle##. Now, let's say for example this state ##\alpha## is an Hamiltonian eigenfunction...
  14. Salmone

    I Separable Hamiltonian for central potential

    In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...
  15. gremory

    A Power series in quantum mechanics

    Just earlier today i was practicing solving some ODEs with the power series method and when i did it to the infinite square well i noticed that my final answer for ##\psi(x)## wouldn't give me the quantised energies. My solution was $$\psi(x) = \sum^{\infty}_{n=0} k^{2n}(\cos(x) + \sin(x))$$...
  16. Lynch101

    B Statistical Independence in Quantum Mechanics

    Very basic question here, about statistical independence in quantum mechanical experiments. The quote from PD below is what prompted the question. When we talk about "some kind of pre-existing correlation" are talking about a simple correlation in the sense of the correlation of sunglasses and...
  17. warhammer

    Cracking Exams: Overcoming Knowledge Gaps for Intro QM

    Summary:: I understand the consensus on PF about studying for knowledge and not merely for "cracking Semester Exams" but I urge you all to go through below thread before attaching to that feeling in my case. Hi. So I have my Exams on Intro QM approaching very soon, which will be a combination...
  18. warhammer

    [Intro QM] Verification requested on possible solution for a question

    Below I have attached an image of my possible solution. I have replaced all the relevant limits. For some reason, I am getting the final value for (i) part as ψ(x)= with an additional √2pi in the denominator. Have I made any errors or is it fine if I take it within the constant A.. In...
  19. Salmone

    I Particle on a cylinder with harmonic oscillator along z-axis

    I need to know if I have solved the following problem well: A spin-less particle of mass m is confined to move on the surface of a cylinder of infinite height with a harmonic potential on the z-axis and Hamiltonian ##H=\frac{p_z^2}{2m}+\frac{L_z^2}{2mR^2}+\frac{1}{2}m\omega^2z^2## and I need to...
  20. P

    A Position basis in Quantum Mechanics

    Can I conceive a countable position basis in Quantum Mechanics? How can I talk about the position basis in the separable Hilbert space?
  21. mohamed_a

    I Classical analogy approach to quantum mechanics

    I have read about several approcahes to bypass some classical restrictions to quantum facts such as the electron being in a torus-like shape to avoid ,the greater than speed of light, rotation paradox . Could you recommend websites , sources or books that give good classical analogy to quantum...
  22. K

    I Particle on a sphere problem in quantum mechanics and its solution

    To solve a particle on a sphere problem in quantum mechanics we get the below equation :##\left[\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\right)-\frac{m^{2}}{\sin ^{2} \theta}\right] \Theta(\theta)=-A \Theta(\theta) ## To solve this differential equation, we...
  23. Mr_Allod

    Position expectation value of 2D harmonic oscillator in magnetic field

    Hello there, for the above problem the wavefunctions can be shown to be: $$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$ Here ##b = \sqrt{1 +...
  24. P

    Commutation relations between Ladder operators and Spherical Harmonics

    I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?
  25. Salmone

    I Inner products with spherical harmonics in quantum mechanics

    Let ##|l,m\rangle## be a simultaneous eigenstate of operators ##L^2## and ##L_z## and we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##. It is true that in general ##\langle l,m|cos(\theta)|l,m'\rangle=0## ##(1)## for the same ##l## even if...
  26. D

    I Ideas about observing position and momentum at the same time

    I am very interested in quantum mechanics/physics and i keep seeing the Heisenberg uncertainty principle and its making me think about other forms of viewing particles. We traditionally use Photons to view something (our eyes), or other forms of radiation/particles, but i know that merely...
  27. thedubdude

    What can a retired electrical engineer learn about physics at 69 years old?

    I join as a 69 year old retired electrical engineer who is interested in physics. I have particular interest in particle physics and quantum mechanics. I don't expect to provide answers on this forum, but I do intend to ask questions.
  28. thedubdude

    B Special Relativity violation via Quantum Mechanics?

    We know that both momentum and position can not be known precisely simultaneously. The more precisely momentum is known means position is more uncertain. In fact, as I understand quantum mechanics, position probability never extends to 0% anywhere in the universe (except at infinity) for any...
  29. F

    I How to define expectation value in relativistic quantum mechanics?

    In non relativistic quantum mechanics, the expectation value of an operator ##\hat{O}## in state ##\psi## is defined as $$<\psi |\hat{O}|\psi>=\int\psi^* \hat{O} \psi dx$$. Since the scalar product in relativistic quantum has been altered into $$|\psi|^2=i\int\left(\psi^*\frac{\partial...
  30. Dario56

    I Kinetic Energy and Potential Energy of Electrons

    Time indepedendent Schrödinger equation for a system (atom or molecule) consisting of N electrons can be written as (with applying Born - Oppenheimer approximation): $$ [(\sum_{i=1}^N - \frac {h^2} {2m} \nabla _i ^2) + \sum_{i=1}^N V(r_i) + \sum_{i < j}^N U(r_i,r_j)] \Psi = E \Psi $$ Terms in...
  31. entropy1

    I Quantum Mechanics without time?

    Is there a view in quantummechanics, of quantummechanics, without time as a concept?
  32. M

    I Informational Interpretation of quantum mechanics?

    I heard something today about the "informational interpretation" of quantum mechanics and a phrase used was "it from bit." Is there actually such a thing? What does it mean, and how is it distinguished from other interpretations like MWI or Copenhagen?
  33. Mr_Allod

    Probability of finding a pion in a small volume of pionic hydrogen

    Hello, I am trying to figure out the right way to approach this. First of all, other than the different Bohr radius value, does the change to a negative pion make any other difference to calculating the probability? Also what would be the correct way to apply the "small volume"? What I'm...
  34. K

    I Definition of magnetic moment in quantum mechanics

    * The general formula for the magnetic moment of a charge configuration is defined as ##\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r##* For an electron it's said that the correct equation relating it's spin and magnetic moment is is ##\vec{\mu} =g\frac{q}{2m}\vec{S}## * It's...
  35. Ravi Mohan

    I am revisiting the mathematical formulation of quantum mechanics with the dimensional (MLT) perspective

    Hi Fellas! My first post after a long hiatus from forums. Feeling nostalgia (this is the place where it all began, my fuel for quantum fascination so to speak). I am revisiting the mathematical formulation of quantum mechanics with the dimensional (MLT) perspective. I want to understand what...
  36. A. Neumaier

    Insights Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

    Continue reading...
  37. M

    Help on Learning Quantum Mechanics (Undergraduate)

    Summary:: I am in the highest level Quantum class at my university- technically considered a grad class. I am an undergrad and need advice on just how to learn it. What study tips? Good Youtubers? Physical simulations? Anything that helped you in quantum mechanics. Hello! I am an undergrad...
  38. Vectronix

    Quantum Modern Quantum Mechanics 3rd Ed: J. J. Sakurai & Jim Napolitano Review

    Is Modern Quantum Mechanics, 3rd Edition, by J. J. Sakurai and Jim Napolitano a good book to learn quantum mechanics from?
  39. S

    I A thought experiment concerning determinism in quantum mechanics

    According to the uncertainty principle, when we measure a micro-object with a measuring device, we cannot predict what value the device will show. But if we knew exactly the wave function of this device, together with the wave function of the micro-object, could we exactly predict the result of...
  40. Santiago24

    Quantum Introductory quantum mechanics textbook for self-study

    Hi! I want to self study some of quantum mechanics so i need introductory textbook. I've taken courses on linear algebra that covers all "Linear algebra done right" by Sheldon Axler, multivariable calculus, two courses on general physics and the basics of differentials equations. I really like...
  41. K

    Quantum Finding the Perfect Self-Study Book for Intro Stats & Quantum Mechanics

    Can you please suggest a good introductory statistical and quantum mechanics book which can be self studied. My math background : I've done multivariate calculus, vector calculus, linear algebra ,some complex analysis all at the usual undergraduate level. The books I've self studied thus far...
  42. tworitdash

    A Applications of weak measurement of quantum mechanics in other domains

    This is a surface level question and I don't want to go into detail. Imagine an algorithm which when used with a sensor output gives the statistical moments of a variable in nature (for example mean and standard deviation of a variable). The sensor measures this once in a while (like once in a...
  43. Shreya

    Phase factor in quantum mechanics

    Please be kind to help.I would be grateful
  44. Hamiltonian

    Quantum Buying my first Quantum mechanics book

    I recently started studying some quantum mechanics, so far I have been using online resources(like MIT OCW 8.04/8.05, and Tongs notes I think I have reached a stage where I understand the Schrodinger eqn and can solve it for various potentials(including for the H-atom) but I don't know anything...
  45. kbansal

    How to explain the Quantum Mechanics/Math of the stages of MRI imaging

    "B0 is a static magnetic field (produced by a superconducting magnet) that initially causes the protons in the body to align with the field and precess at the larmor frequency along the z axis . From a mathematical perspective this precession around the B0 axis occurs due to the time evolution...
  46. J

    A Quantum Field theory vs. many-body Quantum Mechanics

    A lot of people say that Quantum Field theory (QFT) an Quantum Mechanics (QM) are equivalent. Yet, I've found others who dispute these claims. Among the counter-arguments (which I admittedly do not have the expertise to pick apart and check their validity in full) are the following: 1) While QFT...
  47. A. Neumaier

    I Quantum mechanics via quantum tomography

    I just finished a new paper, A. Neumaier, Quantum mechanics via quantum tomography, arXiv:2110.05294. (later renamed to) A. Neumaier, Quantum tomography explains quantum mechanics, arXiv:2110.05294. Abstract: Starting from first principles inspired by quantum tomography rather than Born's...
  48. D

    I Physical interpretation of phase in solutions to Schrodinger's Eqn?

    Hello all, So I've been working through the solutions to some simple introductory problems for the Schrodinger Equation like the infinite square well, and I'm trying to make sense of how to think about the phase component. For simplicity's sake, let's start off by assuming we've measured an...
  49. Morbert

    A Retrodictive Inferences in Quantum Mechanics

    Take a simple case: A system is prepared in state ##\rho_i## at time ##t_0##, and a projective measurement is performed at time ##t_2## with an outcome ##b##. We can retrodict a projective measurement outcome ##a## at time ##t_1## where ##t_0<t_1<t_2##$$p(a|b) =...
  50. A

    I The definition of the spectra in quantum mechanics

    for undergraduate students how to explain this?
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