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. J

    Geometry Book on Differential Geometry/Topology with applications

    Hello! I want to learn about the mathematics of General Relativity, about Topology and Differential Geometry in general. I am looking for a book that has applications in physics. But, most importantly, i want a book that offers geometrical intuition(graphs and illustrations are a huge plus) but...
  2. F

    I Difference between superposition and mixed state

    I get that a if we have complete information of the state of the system (i.e. all the possible knowledge we could have about it: the values its observables can take and their corresponding probabilities), then it is a pure state and can be represented by a vector (ket), ##\lvert\psi\rangle## in...
  3. L

    Similarity Transformation Involving Operators

    Homework Statement Virtually all quantum mechanical calculations involving the harmonic oscillator can be done in terms of the creation and destruction operators and by satisfying the commutation relation \left[a,a^{\dagger}\right] = 1 (A) Compute the similarity transformation...
  4. Perturbative

    I General Relativity within the confines of a Hilbert Space

    Introduction If Quantum Mechanics is more fundamental than General Relativity as most Physicists believe, and Quantum Mechanics is described using Hilbert Spaces wouldn't finding a compatible version of General Relativity that operates within the confines of a Hilbert Space be of utmost...
  5. L

    Other Research topics in Statistical Physics

    Currently I'm in the last year of the Physics course and I've been thinking about working in a project of undergraduate research, specifically in Statistical Physics. Two years ago I've already done a project like that in Fluid Mechanics combined with Gauge Theories and in that project I've...
  6. L

    I Tunnel Effect in Quantum Mechanics: Exploring Time-Independent Problems

    In case of tunnel effect in quantum mechanics we often consider time independent Schroedinger equation with potential ##0##, when ##x<0## then some ##V_0## when ##0\leq x\leq a## and ##0## when ##x>a## so potential barrier problem. And energy of particle that we send to barrier is ##E<V_0##. In...
  7. Kris Lemkens

    B Exploring Quantum Mechanics: Could Another Big Bang Occur in My Hand?

    Just watched the documentary 'Stephen Hawking Grand Design: Did God Create The Universe'. S.H. stated that the big bang was similar to subatomic particles that appear, disappear and reappear somewhere else following the laws of quantum mechanics. If that were so, wouldn't it also be possible for...
  8. J

    I Probability current in positive finite square potential

    Hello! I want to prove that the probability current is a continuous entity at the boundaries of the square for the situation of 0< E< Vo in the problem where V is zero except a finite region in space where it is +Vo and we consider an incoming particle from the left(for example). I thought that...
  9. A. Neumaier

    Insights Misconceptions about Virtual Particles - Comments

    A. Neumaier submitted a new PF Insights post Misconceptions about Virtual Particles Continue reading the Original PF Insights Post.
  10. Kara386

    I Understanding Bra-Ket Notation in Quantum Mechanics

    In my lecture notes, it says that ##\left\langle l \right| A_{nm} \left| \psi \right\rangle = \sum_{n,m} A_{nm} \left\langle m \right| \left|\psi \right\rangle \left\langle l \right| \left| n \right\rangle## ##=\sum_{n,m} A_{nm}\left\langle m \right| \left| \psi \right\rangle \delta_{ln}## ##=...
  11. M

    A Hamiltonian and constant potential

    H=p^2/2m+c What's c? It's of course a shift in energy, but can be thought also as a smoother and smoother real-space local potential that becomes a constant all over the space. On the other hand, why couldn't one think about it as a constant potential in reciprocal space? It's a shift in energy...
  12. G

    Diagonalizing a polynomial of operators (Quantum Mechanics)

    The problem asks for the diagonalization of (a(p^2)+b(x^2))^n, where x and p are position and momentum operators with the commutation relation [x,p]=ihbar. a and b are real on-zero numbers and n is a positive non-zero integer.I know that it is not a good way to use the matrix diagonalization...
  13. jfizzix

    Insights How Does the EPR-Paradox Relate to Entanglement and Nonlocality?

    jfizzix submitted a new PF Insights post Steering: How the EPR-Paradox Fits Between Entanglement and Nonlocality Continue reading the Original PF Insights Post.
  14. J

    Angular momentum of hydrogen atom with Schrodinger Equation

    If we were to assume that the electron moves around the proton with radius a, the Schrodinger equation becomes: ##\frac{1}{a^2}\frac{d^2\psi}{d\phi^2} + \frac{2m}{\hbar^2}|E|\psi = 0## The question in my textbook asks me to solve the above equation to obtain values of energy and angular...
  15. phys-student

    What are the Eigenvalues and Eigenkets of a Spin-1/2 System in a Magnetic Field?

    Homework Statement Consider a spin system with noninteracting spin 1/2 particles. The magnetic moment of the system is written as: μ = (ħq/2mc)σ Where σ = (σx, σy, σz) is the Pauli spin operator of the particle. A magnetic field of strength Bz is applied along the z direction and a second...
  16. Peter25samaha

    Entropy or Quantum Mechanics ?

    is entropy problem easiest than quantum mechanics problem which one is more complicated to understand and to solve ?
  17. Peter25samaha

    What should i study for Quantum Mechanics

    I want to start learning quantum mechanics but i have no idea where to start . I want to know which other fileds and physics branches i have to study before quantum mechanics and if i want to go more deeply for this (like taking a graduate course ) what should i study before this in physics math...
  18. M

    Quantum Mechanics (finding the Hamiltonian of a quantum top)

    Hello, This was part of my midterm exam that i couldn't solve. Any help is extremely appreciated. Problem: The K.E. of a rotating top is given as L^2/2I where L is its angular momentum and I is its moment of inertia. Consider a charged top placed at a constant magnetic field. Assume that the...
  19. T

    A What do people mean by "dynamical phase transition"?

    I heard of it several times in colloquiums. But I still cannot grasp the meaning. What is the control parameter in this phase transition?
  20. Amrator

    Other What Exactly is an Introductory Level for Quantum Mechanics?

    I recently bought this book: https://www.amazon.com/dp/1107050405/?tag=pfamazon01-20 In the preface it says that I'm assumed to have already learned quantum mechanics (including angular momentum and Dirac equation) and nuclear physics at an introductory level. What does that mean? Do they mean...
  21. marcus

    A Weinberg: Quantum Mechanics Without State Vectors

    Steven Weinberg has made what he calls a "modest proposal": http://arxiv.org/abs/1405.3483 Quantum Mechanics Without State Vectors Steven Weinberg (Submitted on 14 May 2014) It is proposed to give up the description of physical states in terms of ensembles of state vectors with various...
  22. Ryaners

    Electron in 1-D box: photon absorbed?

    I don't know where I'm going wrong with this problem - I was so sure I had it right but the online grader tells me otherwise :oldfrown: Homework Statement An electron in a one-dimensional box has ground-state energy 2.60 eV. What is the wavelength of the photon absorbed when the electron...
  23. KarminValso1724

    B Does quantum mechanics explain why subatomic particles behave the way they do?

    For example, general relativity relates the behavior of gravity the the deformation of spadetime. But does quantum mechanics explain why particles behave the way they do? Or does it only explain how processes such as entanglement work not why they occur.
  24. DrPapper

    I Expression for Uncertainty of Arbitrary Operator

    Hello all, as far as I can see this question is not posted already, my apologies if it is and please provide a link. But I'm watching this video on youtube: And at 22:38 there's an expression given for the uncertainty of an arbitrary operator Q, however I'm concerned the expression is incorrect...
  25. E

    I Energy of a number of particles

    Hello! It is sometimes useful to find the average energy of a certain number N of particles contained in a box of volume V. In order to find this quantity, the total energy is required and then divided by N. The result is E_{average} = \displaystyle \frac{1}{N} \sum_{n = 1}^{N} \left| a_n...
  26. thegirl

    I Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

    Hi, why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity. I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and...
  27. michaelmolli

    Particle in One-Dimensional Box Problem [Quantum Mechanics]

    Homework Statement a) Determine the ratio (Em/En) between two energy states of a particle in a one dimensional box of length l. b) Show that this is consistent with the non-relativistic low-energy limit. The attempt at a solution I have figured out a) using the de broglie wave-particle duality...
  28. F

    Quantum Mechanics: Delta Potential Sudden Change

    Homework Statement I have a particle which is initially in a bound state for a given voltage in the form of a delta function at the origin, V = -αδ(x) initial state is ψα = (√αm)/h2*exp(-m*α*|x|/(h2) At t=0, voltage is changed to V = -βδ(x) both α and β are greater than zero. Right now I'm...
  29. Schwarzschild90

    Transfer matrix for a finite length? (Quantum mechanics)

    Homework Statement I'm struggling to find a solution to exercise (*b). I have uploaded a pdf of the assignment. Please advise me at your convenience. Homework Equations x(x_l^+) = T(x_l^+, x_l^-)x(x_l^-) The Attempt at a Solution x(a^-) = \frac{\psi(a^-)}{\psi(a^-)} , T(a^+, a^-) \left(...
  30. J

    I How to prove Schrodinger's equation in momentum space?

    Specifically, i do not know hot to express the potential in momentum space. If someone would provide me with a link of source that has the proof in it, it would be appreciated.
  31. A. Neumaier

    A Effective Dynamics of Open Quantum Systems: Stochastic vs Unitary Models

    Not quite. But it necessarily has to be described by a different quantum model than unitary dynamics if it is an open system and the rest of the universe is not explicitly modeled. For convenience, physicists often want to describe a small quantum system in terms of only its Hilbert space, when...
  32. Danny Boy

    I Why is reflection coefficient defined this way

    In Griffith's "Introduction to Quantum Mechanics, second edition" he states: For the delta-function potential, when considering the scattered states (with E > 0), we have the general solutions for the time-independent Schrodinger equation: $$\psi(x) = Ae^{ikx} + Be^{-ikx}~~~~\text{for }x<0$$ and...
  33. E

    Free Particle moving in one dimension problem

    Homework Statement 5) A free particle moving in one dimension is in the state Ψ(x) = ∫ isin(ak)e(−(ak)2/2)e(ikx) dk a) What values of momentum will not be found? b) If the momentum of the particle in this state is measured, in which momentum state is the particle most likely to be found? c)...
  34. Imran Makhdoom

    B Quantum mechanics and reality of matter

    Hi Do Quantum Mechanics negates the reality of matter and proves un-reality of matter ?
  35. F

    Why do we need Quantum Mechanics so much?

    Almost phenomena at macro level are classical phenomena,why physics needs QM very much?Is it rational need or practical need?
  36. M

    Quantum non-locality and vacuum polarization

    Quantum particles are not localized before they are observed, as shown with the Young double slit experiments and those with entangled particles. On the other side, vacuum is filled with virtual particles. Are the non-localized particles responsible for the virtual particles? or only for a part...
  37. aleazk

    Tools to enrich our quantum mechanics interpretations discourse - Comments

    aleazk submitted a new PF Insights post Tools to Enrich our Quantum Mechanics Interpretations Discourse Continue reading the Original PF Insights Post.
  38. C

    B Quantum Mechanics vs. Pilot-Wave thought experiment

    Hello. I want to share a thought experiment that could tell Quantum Mechanics apart from Pilot-Wave interpretation. It goes like this: Quantum Mechanics vs. Pilot-Wave: Quantum Mechanics: Waves collapse to particles. Waves disappear when particles are detected. Pilot-Wave: Waves are real but...
  39. G

    Harmonic oscillator positive position expectation value?

    So this is something that troubled me a bit- in Shankar's PQM, there's an exercise that asks you to find the position expectation value for the harmonic oscillator in a state \psi such that \psi=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle) Where |n\rangle is the n^{th} energy eigenstate of...
  40. T

    Why is the electromagnetic field not a 'charge field'

    This question is a continuation/topic-extrapolation of: https://www.physicsforums.com/threads/flux-in-magnetic-core-according-to-special-relativity.856482/#post-5374651 My question is 'how is the electromagnetic field different from some sort of mere electric-charge field?' The issue I have...
  41. omidaut

    Advanced quantum mechanics book with solutions

    Hi there! Can you please introduce me some books on advanced quantum mechanics which has solutions for its exercises. Of course, I know Shankar and Sakuraii, but, I mean something more advanced which covers these topics: (Scattering, Dirac Fields, Group Representations, Relativistic Quantum...
  42. DaTario

    A Amplitudes of probability in Mathematics before QM

    Hi All, Was there any use of the concept of amplitudes of probability before their use in quantum mechanics? In connection to this question, who invented or was the first to use this resource? Best wishes, DaTario
  43. C

    How do I evaluate <x> with the k-space representation?

    Homework Statement Given the following k-space representation of the wave function: Ψ(k,t) = Ψ(k)e-iħk2t/2m use the wave number representation to show the following: <x>t=<x>0 + <p>0t/m <p>t=<p>0 Homework Equations <x>=∫Ψ*(x,t)xΨ(x,t)dx <p>=∫Ψ*(x,t)(-iħ ∂/∂x)Ψ(x,t)dx The Attempt at a...
  44. smodak

    Quantum Spin First Approach to Quantum Mechanics Textbook

    So far I have seen Sakurai, Townsend, Cohen Tannoudji, Feynman, and McIntyre. Are there any other books that take this approach? Just curious.
  45. W

    Berry's Curvature Equation cross product calculation

    Hi, The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation Vm= (- 1/B2 ) * i *∑ ( <m,B|S|n,B> ∧ <n,B|S|m,B> ) / A2 ...[1] the textbook claims that we add the term m = n since <m|S|m> ∧ <m|S|m>...
  46. M

    Quantum Ballentine's book: 1st or 2nd edition?

    I'm planning to get a copy of Quantum Mechanics - Modern Development by L. E. Ballentine. However I am uncertain between the first and second editions. The first edition is cheaper. I will be using it for my PhD research with the topic of atomic physics. Will the second edition give me...
  47. L

    Finding the State of a Quantum System with Given Hamiltonian and Observable

    Homework Statement We are given the Hamiltonian H and an observable A ##H=\begin{pmatrix} 2 & 0 & 0\\0 & 0 & 1\\0 & 1 & 0 \end{pmatrix}\hbar\omega A= \begin{pmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -1 \end{pmatrix}a ## We are also told that at ##t=0## we have that a measurement of A gives us...
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