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. LarryC

    Simultaneous Diagonalization for Two Self-Adjoint Operators

    (a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?
  2. Sushmita

    A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x)....

    Homework Statement [/B] A particle of mass 'm' is initially in a ground state of 1- D Harmonic oscillator potential V(x) = (1/2) kx2 . If the spring constant of the oscillator is suddenly doubled, then the probability of finding the particle in ground state of new potential will be? (A)...
  3. J

    Quantum I need a book to solve Schrodinger's eqn numerically

    I have found this one that looks perfect: https://www.amazon.com/dp/331999929X/?tag=pfamazon01-20 THe problem is that it has not been published yet :( , but I can't believe there is no other book on the subject. What I want is to solve numerically the Schrodinger equation with no special...
  4. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...
  5. D

    Other What books to get before studying Quantum Mechanics and General Relativity?

    Hi. I want to learn - amateurishly - Quantum Mechanics, and General Relativity, but my experience with Physics is very small. I want to ask, what should I learn - what books should I read - before I start to learn those theories? Sorry for my english.
  6. DrClaude

    Two-level quantum system (from Sakurai)

    Homework Statement Sakurai, problem 1.11 A two-state system is characterized by the Hamiltonian $$ H = H_{11} | 1 \rangle \langle 1| + H_{22} | 2 \rangle \langle 2| + H_{12} \left[ | 1 \rangle \langle 2| + | 2 \rangle \langle 1| \right] $$ where ##H_{11}##, ##H_{22}##, and ##H_{12}## are real...
  7. smodak

    Quantum New introductory Quantum Mechanics book from Schwichtenberg

    I liked his Group Theory book. Now he wrote a QM book. No-Nonsense Quantum Mechanics: The Ultimate No Holds Barred Guide To The Quantum World
  8. CharlieCW

    Is \(|p,\lambda\rangle\) an Eigenstate of the Helicity Operator?

    Homework Statement For massless particles, we can take as reference the vector ##p^{\mu}_R=(1,0,0,1)## and note that any vector ##p## can be written as ##p^{\mu}=L(p)^{\mu}_{\nu}p^{\nu}_R##, where ##L(p)## is the Lorentz transform of the form $$L(p)=exp(i\phi J^{(21)})exp(i\theta...
  9. CharlieCW

    Transforming one matrix base to another

    Homework Statement The SO(3) representation can be represented as ##3\times 3## matrices with the following form: $$J_1=\frac{1}{\sqrt{2}}\left(\matrix{0&1&0\\1&0&1\\ 0&1&0}\right) \ \ ; \ \ J_2=\frac{1}{\sqrt{2}}\left(\matrix{0&-i&0\\i&0&-i\\ 0&i&0}\right) \ \ ; \ \...
  10. Pushoam

    Momentum measurement of a particle in Quantum Mechanics

    Homework Statement What will momentum measurement of a particle whose wave - function is given by ## \psi = e^{i3x} + 2e^{ix} ## yield? Sketch the probability distribution of finding the particle between x = 0 to x = 2π. Homework Equations The Attempt at a Solution The eigenfunctions of...
  11. S

    I Understanding Spin & Angular Momentum in Quantum Mechanics

    Hello! I got a bit confused about the fact that the whole the description of spin (and angular momentum) is done in the z direction. So, if we are told that a system of 2 particles is in a singlet state i.e. $$\frac{\uparrow \downarrow -\downarrow \uparrow }{2}$$ does this mean that measuring...
  12. P

    Stimulated emission per second per atom?

    Homework Statement We are investigating hydrogen in a plasma with the temperature 4500 ºC. Calculate the probability per atom and second for stimulated emission from 2p to 1s if the lifetime of 2p is 1.6 ns Homework Equations Planks radiation law: ##\rho (f) = \frac{8* \pi...
  13. I

    How is the motion of an electron around nucleus?

    How electron revolves around orbitals? Is the velocity uniform or accelerated?
  14. CharlieCW

    Coherent states in quantum mechanics (Schrödinger Cat)

    Hello. I've been struggling for a day with the following problem on Quantum coherent states, so I was wondering if you could tell me if I'm going in the right direction (I've read the books of Sakurai and Weinberg but can't seem to find an answer) 1. Homework Statement *Suppose a Schrödinger...
  15. D

    I Finding expansion coefficient of a 3-d Gaussian wave packet

    I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet. First I'm decomposing a Gaussian wave packet $$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0...
  16. D

    I How Do Quantum Field Theory and Quantum Loop Gravity Explain the Quantum World?

    I think this could be an interesting discussion (unless I'm just totally off base): I don’t think were looking at things right in the atomic world. We represent everything as this wave particle duality which is not incorrect but is a great way to visualize particles and forces and their...
  17. B

    Schwinger's model of angular momentum

    Homework Statement Consider two pairs of operators Xα, Pα, with α=1,2, that satisfy the commutation relationships [Xα,Pβ]=ihδαβ,[Xα,Xβ]=0,[Pα,Pβ]=0. These are two copies of the canonical algebra of the phase space. a) Define the operators $$a_\alpha =...
  18. M

    Quantum First edition of Ballentine's Quantum Mechanics book

    Hi. I just received a copy of Ballentine's "Quantum Mechanics: A Modern Development" ordered from Amazon, after I heard well of it in this site. I'm wondering what edition I've bought: the one I've got has a white hardcover, and its ISBN is 9789810227074. Does someone ever used this edition? Why...
  19. D

    Quantum Mechanics, What is it exactly?

    I recently watched a video where Sean Carroll talked about QM and multiverses 1/ Could you please explain: Where does all the energy come from to drive all these universes. Surely this must take an enormous amount of energy to drive a multiverse system (infinite). Does each universe...
  20. RUTA

    Insights Why the Quantum | A Response to Wheeler's 1986 Paper - Comments

    Greg Bernhardt submitted a new PF Insights post Why the Quantum | A Response to Wheeler's 1986 Paper Continue reading the Original PF Insights Post.
  21. Wrichik Basu

    Courses What is meant by "basic" quantum mechanics?

    I have been taking many online courses. In the prerequisites for many courses, it has been mentioned, "basic" quantum mechanics. It has become important to define where the boundary of basic ends and the advanced level starts, though I believe that is not well defined. I have been studying QM...
  22. P

    Question about Many Worlds branching in Quantum Mechanics

    Supposing the Many Worlds interpretation of QM is true... If a branching occurs during what we perceive is a wave function collapse, why would this be perceptible to us as probabilties? Wouldn't we just branch, leaving it just as imperceviable as the passage of time? That is, it just happens...
  23. thariya

    I The sign of coupling Hamiltonian in CQED

    Hi all, I've always regarded the coupling Hamiltonian for a bosonic cavity mode coupled to a two-level fermionic gain medium chromophore to be of the form, $$H_{coupling}=\hbar g(\sigma_{10}+\sigma_{01})(b+b^{\dagger})$$, where ##b## and ##b^{\dagger}## and annihilation and creation operators...
  24. Don

    Popular book on Quantum mechanics

    I lost my book on Quantum mechanics! It was published in the late 80s or early 90s in England. Title: Quantum Mechanics. The book used algebra with more advanced math in the appendices. If you know the author , please reply. Thanks!
  25. H

    Quantum Exploring Quantum Mechanics: ER=EPR, Black Holes, Firewalls & More

    ER=EPR, black hole complementary, firewalls, vacuum entanglement etc.. Where do I begin studying these new ideas? I have a solid understanding about Quantum Field Theory and the classical theory of gravity, but no knowledge of string theory. Are there some advice or book recommendations anyone...
  26. C

    I Particle Physics and Quantum Mechanics

    I’m having a hard time, as I begin learning QM, knowing what it applies to, if I can put it that way. Is QM the rules that describe how the particles of the Standard Model interact with each other? Or what is the best way to understand the relationship between what one studies when one studies...
  27. S

    A Can quantum cellular automata simulate quantum continuous processes?

    Can quantum cellular automata/quantum game of life simulate quantum continuous processes in the continuous limit? At the end of this article: https://hal.archives-ouvertes.fr/hal-00542373/document it is said that: "For example, several works simulate quantum field theoretical equations in the...
  28. gmalcolm77

    Where Can I Find Free Online Courses on Quantum Mechanics?

    :rolleyes: I would like to find a free online course, not too hard. I have minors in math and physics, but have been away for awhile. Maybe something on youtube. If someone knows of a decent course that I could educate myself with I would appreciate any info on it. Thanks in advance...
  29. K

    I Interval in Quantum Mechanics?

    In Special/General Relativity invariance of a space-time interval is just so important. But in Quantum Mechanics, be it non-relativistic or QFT, there seems to be no such parallel. I have always noticed this. I have some ideas about the reason: 1 - it's not part of the theory to have a...
  30. E

    First order perturbation energy correction to H-like atom

    Homework Statement Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential $$ \phi(r) = \begin{cases} \frac{Ze}{R}(\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}) &\quad r<R\\ \frac{Ze}{r} &\quad r>R \\...
  31. anorlunda

    Insights How to Better Define Information in Physics - Comments

    Greg Bernhardt submitted a new PF Insights post How to Better Define Information in Physics Continue reading the Original PF Insights Post.
  32. A

    A Rigorous transition from discrete to continuous basis

    Hi all, I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of $$\left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left...
  33. FourEyedRaven

    Quantum What's wrong with the Messiah "Quantum Mechanics" textbook?

    Hi. I bought Messiah's "Quantum Mechanics" because it was at an excelent price from Dover. But, even though it was considered a Bible of quantum mechanics until recently, people consider it outdated now. Is it no longer comprehensive? I intend to read on relativistic quantum mechanics and...
  34. Safder Aree

    Wave packet width given a wave function

    Homework Statement Find the wave packet Ψ(x, t) if φ(k) = A for k0 − ∆k ≤ k ≤ k0 + ∆k and φ(k) = 0 for all other k. The system’s dispersion relation is ω = vk, where v is a constant. What is the wave packet’s width? Homework Equations [/B] I solved for Ψ(x, t): $$\Psi(x,t) =...
  35. S

    I What exactly is Weizsäcker's ur-alternatives theory?

    What is exactly Weizsäcker's ur-alternatives theory? How is it related to digital physics theories? Is it related to pancomputationalism? Does it defend that a universe can be described as being fundamentally made of qubits? Would this mean that that universe would be fundamentally made by...
  36. Safder Aree

    Harmonic Oscillator violating Heisenberg's Uncertainity

    Homework Statement Does the n = 2 state of a quantum harmonic oscillator violate the Heisenberg Uncertainty Principle? Homework Equations $$\sigma_x\sigma_p = \frac{\hbar}{2}$$ The Attempt at a Solution [/B] I worked out the solution for the second state of the harmonic oscillator...
  37. A

    I Does the de Broglie-Bohm pilot wave theory make predictions?

    I find the de Broglie-Bohm pilot wave theory interesting but what I still feel missing in the descriptions I could find so far is that it reformulates what we already know but nobody speaks of new testable predictions that could eventually distinguish it from other interpretations (such as a new...
  38. WeiShan Ng

    I Momentum/Position space wave function

    These are from Griffith's: My lecture note says that I am having quite a confusion over here...Does the ##\Psi## in the expression ##\langle f_p|\Psi \rangle## equals to ##\Psi(x,t)##? I understand it as ##\Psi(x,t)## being the component of the position basis to form ##\Psi##, so...
  39. Jarvis323

    I Definition and Rules of Quantum State Observation

    I was wondering how the rules work for observation in a quantum system. Particularly, about what happens if two separate entities try measuring at the same time. And also, what kinds of interactions are happening all the time that are considered measurements, for example in quantum...
  40. CDL

    Determining a Scattering Cross Section (Quantum Mechanics)

    Homework Statement Consider scattering of a particle of mass ##m## on the potential $$U(r) = \begin{cases} 0, & r \geq b\\ W, & r < b \\ \end{cases}$$ Where ##W## is some arbitrary chosen constant, and the radius ##b## is considered a small parameter. Find the cross section ##\sigma## in the...
  41. J

    Measuring momentum using position wavefunction

    I was solving an exercise from Cohen's textbook, but then I got stuck in this question. "Let ψ(x,y,z) = ψ(r) the normalized wave function of a particle. Express in terms of ψ(r) the probability for: b) a measurement of the component Px of the momentum, to yield a result included between p1...
  42. M

    Stuck on Quantum Mechanics Potential Steps Problem?

    Homework Statement The Attempt at a Solution [/B] Hi All, I'm having trouble answering part (f) of the above question. I have managed parts (d) and (e) fine but am not sure how to proceed with part (f). I am pretty sure that the amplitude of the reflected wave in region 1 will be zero...
  43. G

    Why is θ Limited to π/2 in Basis Choice for Distinct States?

    Homework Statement Have to read a paper and somewhere along the line it claims that for any distinct ## \ket{\phi_{0}}## and ##\ket{\phi_{1}}## we can choose a basis s.t. ## \ket{\phi_{0}}= \cos\frac{\theta}{2}\ket{0} + \sin\frac{\theta}{2}\ket{1}, \hspace{0.5cm} \ket{\phi_{1}}=...
  44. J

    A Does the Frauchiger-Renner Theorem prove only MWI is correct

    Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page. I am asking in reference to this paper: https://arxiv.org/pdf/1604.07422.pdf Which claims to show that single...
  45. K

    B Lack of Numerical Examples in Relativity & Quantum Mechanics: Exploring Reasons

    It's rare to encounter concrete, numerical examples of what is being taught about Relativity, Quantum Mechanics.. On the other hand there's plenty of numerical examples in the undergraduate general physics textbooks, for instance problems of mechanics. As for General Relativity I did find only...
  46. Riotto

    I Perturbative versus nonperturbative quantum mechanics

    What is the nonperturbative approach to quantum mechanics as opposed to perturbative one? When does the latter method fail and one has to apply nonperturbative approach? Please keep your discussion confined within non-relativistic quantum mechanics.
  47. A. Neumaier

    I Relativistic hidden variable quantum mechanics?

    I just came across the following paper: Gisin, N. (2011). Impossibility of covariant deterministic nonlocal hidden-variable extensions of quantum theory. Physical Review A, 83(2), 020102. proving that, under sensible hypotheses, nonlocal hidden variable theories for relativistic quantum...
  48. Semiconductor

    B Unveiling the Mystery of Invisible Atoms

    If atom are invisible then how can one say that atoms are made up of electrons nucleus and quasi particles.
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