Superposition Definition and 66 Discussions

The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y).
A function


{\displaystyle F(x)}
that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity














{\displaystyle F(x_{1}+x_{2})=F(x_{1})+F(x_{2})\,}


{\displaystyle F(ax)=aF(x)\,}
Homogeneityfor scalar a.This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier and Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behavior.
The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. If the superposition holds, then it automatically also holds for all linear operations applied on these functions (due to definition), such as gradients, differentials or integrals (if they exist).

View More On
  1. R

    B Does the bomb experiment do a measurement?

    In short, the Elitzur–Vaidman bomb experiment consists of a Mach–Zehnder interferometer, where a bomb is placed in one of the paths (I used the wikipedia description, and Sabine Hossenfelder's video: ) The bomb can be live or a...
  2. E

    B Question about quantum superposition

    All of my speculation is based on my current understanding of quantum physics as an art high school student who just has this as an interest, which is in no way at a quantum physicist's level so I apologize if this question is stupid. Also sorry for my English. Most, if not all of you reading...
  3. entropy1

    I Schrödinger's Cat and radioactive atom

    Particles can be made to be in superposition of their states. Concerning Schrödingers cat, if the cat is in superposition of being dead and alive, does that mean that the atom that drives the narcotic is in superposition of having decayed and not having decayed? And does that mean that any...
  4. I

    B The superposition of a photon

    When a photon travels through space it is spreading out like a fan while in its superposition (except that it is spreading also vertically in addition to horizontally). So, what happens if for instance the right outer edge of the photon's superposition is captured by the gravity of a black hole...
  5. S

    Engineering Please help me with this circuit question using the superposition theorem

    I have tried many times to solve this network, but can't understand how to get current in each resistors by superposition theorem. Please help me to solve and find currents in each 3 resistors with solution. Note:- The figure is attached below.
  6. Ali Beladi

    I What does this tricky quantum mechanics equation mean?

    I'm a current high school student and I’m aspiring to become a biochemist. I’m at the moment writing an article about adaptive mutations but there is a lot of tricky quantum mechanics in it which I simply don't get. I have asked everyone and got no answer until someone recommended to ask it in a...
  7. H

    I A statement in a superposition of being true and false?

    If a cat can be in a superposition of being dead and alive, why can't a statement be in a superposition of being true and false?
  8. entropy1

    I Superposition of large objects

    If a buckyball can be in superposition as a whole, why can't bigger objects like (the size of) a human body? Why are macro objects not in superposition (after the due operation)? Or are they? I am not sure if this question has an official answer, but math to illustrate an answer is appreciated...
  9. Marcin

    Double slit experiment with magnetic traps

    1. Is it possible to perform the following, modified double-slit experiment with single electrons/electron beam: The slits are separated in such a way, that each leads to a separate magnetic trap. The traps have oppositely directed magnetic fields, so that the electrons entering them move in...
  10. entropy1

    I Observer superposition in MWI?

    In MWI, would you say that a measurement puts the observer in superposition of being in the various worldlines? If I said "yes" to that, would I be correct?
  11. Quantum Entanglement: Spooky Action at a Distance

    Quantum Entanglement: Spooky Action at a Distance

    Fermilab's Don Lincoln describes Quantum Entanglement, and how it is different from classical statistics and hidden variables scenario.
  12. Lapse

    Comp Sci How do I determine the linear/non-linearity of this problem?

    A system is linear if it satisfies the properties of superposition and homogeneity. Superposition: adding the inputs of two systems results in the addition of the two outputs. Ex) x1(t) + x2(t) = y1(t) + y2(t) Homogeneity: multiplying the input by some scalar value is equal to the output...
  13. J

    Finding the missing current and voltage in a circuit with superposition

    I don't get this. Since we have analyzed the circuit separately for each source, adding them should give me the final values of I1, V2, I2, V1 etc. However, that's not quite true—from cases 1 and 2, I should have I1 = 4 + 5 = 9 mA, but it's 8mA. Hence, I thought that the black box consumes 1mA...
  14. Quantum Alchemy

    I Questions about QFT and the reality of subatomic particles

    I've been reading about Quantum Field Theory and what it says about subatomic particles. I've read that QFT regards particles as excited states of underlying quantum fields. If this is the case, how can particles be regarded as objective? It seems to me that this also removes some of the...
  15. S

    I Did Schrödinger himself take his "Cat Experiment" seriously?

    In 1935, Austrian physicist Erwin Schrödinger was looking at a concept called a "superposition." Superposition is when two waves meet and overlap and interact, which can lead to different results based on the circumstances. The concept can be seen in the regular-sized world as well, in...
  16. entropy1

    I Superposition of branches in MWI

    So I consider a measurement on a superposition, in MWI, leads to another superposition: ##(|A\rangle+|B\rangle)|Observer\rangle \rightarrow |A\rangle|Observer{A}\rangle+|B\rangle|Observer{B}\rangle## If we come to the latter situation, a superposition of branches, why does that not mean that...
  17. entropy1

    I Portrayal of spin measurement

    Would this be an accurate portrayal of measuring the spin of an electron with a SG detector?: The electron is in a superposition of spin-up and spin-down; Upon entering the magnetic field of the SG detector, the electron enters a superposition of an upward trajectory and a downward trajectory...
  18. entropy1

    I In which cases does superposition lead to splitting of worlds?

    I don't know how to understand this: Suppose we measure the spin of an electron with apparatus M. M can yield spin-up or spin-down. According to MWI, M briefly becomes in superposition of measuring spin up and spin down. Extremely quick however, M gets split by the splitting into two universes...
  19. TachyonLord

    Newton's Rings Experiment

    When monochromatic light is incident on a plano convex lens(as shown in the picture), these dark rings are produced which are observed with the help of a traveling microscope. The procedure requires us to measure the diameter of each ring (We need to measure the diameter of at least 10...
  20. LarryS

    I Does wave superposition consume time?

    In Classical Mechanics, waves produced in linear systems, like EM waves, obey the Superposition Principle in which the wave amplitudes of, say two input waves, “add up” to create one output wave whose varying amplitude is the sum of the two input waves. One example would be Young’s Double Slit...
  21. S

    I Superposition analogy?

    Thank you to the very helpful forum leaders for their aide in comprehending entangled particle detection last week. With regards to superposition, would an analogy be: (which involves a thought experiment): a ceiling fan, at rest, has four distinct blades with four known positions...
  22. P

    I Grover algorithm geometric interpretation

    Good day everybody, I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for. In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's...
  23. G

    Sound Wave Interference Problem

    Homework Statement This is just a question about a question in Serway & Jewett's "Physics for Scientists and Engineers 3rd Ed". It's Objective Question 3 from Chapter 18, building on Example 18.1 from the text. Two identical loudspeakers placed 3.00 m apart are driven by the same oscillator...
  24. How to Teleport Schrödinger's Cat using entangled flees - minutephysics

    How to Teleport Schrödinger's Cat using entangled flees - minutephysics

    How to teleport Schrödinger’s cat: this video presents the full quantum teleportation procedure, in which an arbitrary qubit (spin, etc) is teleported from Alice to Bob by way of a pair of particles entangled in a bell (EPR) state and the transmission of information via a classical channel.
  25. learn.steadfast

    I Hermitian and expectation values.... imaginary?

    I've been studying quantum mechanics, and working problems to get a feel for expectation values and what causes them to be real. I was working the problem of finite 1D wells, when I came across a situation I did not understand. A stationary state solution is made up of a forward and reverse...
  26. G

    Are superpositions of waves with different c still waves?

    Hi. As far as I know, superpositions of waves are normally considered to be waves too, even in dispersive media. But how can they still be solutions of a wave equation of the form $$\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\Delta\right)u=0$$ if ##c## isn't the same for all of them...
  27. 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...
  28. 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...
  29. C

    Pressure on a Piston

    Homework Statement Suppose there is a tank filled with water and a piston of area S exerts a force F on the water. Suppose I divide the water boundary touching the piston to - N small equal " square " molecules. Then , the force on the upper face of each molecule is F/N . Also, the area of...
  30. entropy1

    I Do macro objects get entangled?

    If we consider the Unitary evolution of the wavefunction, and interpret measurements as becoming in superposition, taking it that the measurement device gets in a superposition of spin up and spin down, do two measurement devices that each measure one particle of an entangled pair become...
  31. G

    A Kirchhoff problem: Find the currents and voltages in this circuit

    Homework Statement In the circuit of the figure, determine the intensities I1, I2, I3, I4, I5 and the voltage Ug. Homework Equations Superposition and Kirchoff The Attempt at a Solution 20-10I1+5I3-10I1=0 5I3+Ug+5I5=0 20I2+10I2-150=0 From the last equation, 30I2 = 150, I2 = 5 A KNOTS...
  32. AuxPart

    I Does the Schrödinger equation link position and momentum?

    I recently found this article about the dynamics of the wave function. It has some good simple illustrations and I found it valuable. But the author has a question himself, about understanding the Schrodinger equation. I wonder if anybody here could fill in the missing piece. The relevant part...
  33. K

    Conservation of energy of a perfectly decoherent laser

    The energy and momentum of a closed system is always conserved. If a laser can emit perfectly parallel and perfectly superimposed light waves (occupy the exact same space) that are perfectly 180° out of phase and of the same frequency, what can be said about the energy and momentum of the...
  34. Ajaroy

    How to calculate the path difference in this question

    1. Homework Statement Two speakers S1 and S2 derived by the same amplifier and placed at y=1 and y = - 1.The speakers vibrate in phase at 600 Hz. A man stands on the x-axis at a very large distance from the origin and starts moving parallel to the Y - axis. The speed of sound in air is 330ms-1...
  35. entropy1

    Simple question about MWI superposition

    I am sorry to bore you people with a very, very simple question (there is no catch unless QM throws it :nb) ): ?:) If we have a half silvered mirror and we fire a single photon at it at an angle of 45°, it either passes or reflects from the mirror. To be more precise, the photon becomes in a...
  36. Y

    B Is this an accurate description of standing waves?

    Hi all, is my description below a reasonable attempt in explaining how a standing wave forms? The main part I am a bit confused as to how to explain is why the antinodes move up and down. Thanks! A standing wave is formed when energy of a wave of the right frequency is trapped in the system...
  37. S

    Beat frequency question

    Homework Statement Homework Equations ##f_beat=f_1-f_2## The Attempt at a Solution Why are the two different answers? Is it because the first question is asking for how often it fluctuates, and the other is actually asking for the frequency of the sound? Why is the resultant tone the...
  38. Const@ntine

    Wave Superposition

    Homework Statement y1(x,t) = 5.00sin(2.00x - 10.0t) y2(x,t) = 10.0cos(2.00x - 10.0t) a) Prove that the wave that is the result of the superposition is a function of sin. b) What's the phase angle and amplitude of said wave? Homework Equations y = y1 + y2 The Attempt at a Solution...
  39. nomadreid

    I Do quantum fluctuations mean metric fluctuations?

    I suspect the following reasoning is faulty, but I am not sure why. Hence I would appreciate someone pointing out the errors. That is, which, if any, of the following statements are incorrect, and why? 1) Theoretically, albeit not practically due to the large numbers involved, the laws of...
  40. Mzzed

    Superposition's effect on frequency?

    Imagine two electromagnetic waves are traveling in opposing directions such that they eventually meet, overlap, and continue traveling as usual after the superposition/overlapping event. Just before they overlap there is some superposition effect taking place but not enough to fully merge the...
  41. Zahidur

    B Adding two waves with different amplitudes

    Would there be destructive interference if I had two waves that are superimposed and the peak of one wave met the trough of the other but they both have different amplitudes? If there isn't total destructive interference, then what does the final wave look like?
  42. snate

    I Confused about complex numbers

    Can someone please explain what's going on at 47:40 Thanks in advance.
  43. D

    Calculate the electric field using superposition

    Homework Statement Find the expression for the electric field at point M(a,a,0) if the linear charge density is known ( ##Q'## ) Homework Equations 3. The Attempt at a Solution [/B] I tried something like this and would like your feedback on it. I separated the triangle into three parts...
  44. A. Neumaier

    I The typical and the exceptional in physics

    For properly normalized extensive macroscopic properties (and this includes the center of mass operator), there is such a proof in many treatises of statistical mechanics. It is the quantum analogue of the system size expansion for classical stochastic processes. For example, see Theorem 9.3.3...
  45. entropy1

    I The mechanism of entangled particles

    First let me ask this: Consider a pair of entangled photons fired at a respective detector after passing respective polarisation filters. If a photon passes a polarisation filter, is it in a superposition of having passed and not having passed? Is the measuring device (that detects the...
  46. H

    B Still confused about superposition and mixed states

    I am confused about pure state or in mixed states. I've seen several threads on this forum, but I still can't get the grasp of it. I only have very little quantum chemistry to know what these means. So instead, I want to know the answer for specific examples so that I can get an idea. So I...
  47. F

    I Quantum superposition and its physical interpretation

    I understand that if we have a quantum mechanical system, then its state at some given time ##t## is fully described by a state vector ##\lvert\psi(t)\rangle## in a corresponding Hilbert space. This state vector containing all possible information about the distributions (of all possible values)...
  48. C

    Confusion regarding the speed of sound in wind

    One of my students asked me the seemingly innocuous question of "how does wind affect the speed of sound?". My immediate thought was that the velocity of the wave would be the vector sum of the velocity of the wind and the velocity of sound waves in still air. However, upon further reflection I...
  49. 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...
  50. Josephthe2

    Calculating deflection with rotation about center support

    I am preparing for a qualifying exam for my PhD program and am looking at some of the old tests from previous years (as supplied by the school for study/prep material). I have come across a deflection problem that has me stumped, and I might just be overthinking it. The problem: