Questions tagged [born-rule]

The Born rule is a rule in Quantum Mechanics that states that the probability density $\rho$ is $|\psi|^2$ where $\psi$ is the probability amplitude.

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Quantum probabilities of a projection - does it require two different definitions…?

Say I have a wave-function $$ |\psi \rangle=\pmatrix{a_1+ib_1\\a_2+ib_2} $$ where of course $\langle \psi |\psi \rangle=1$. I can get the probability of a given state as follows: $$ \begin{align} \...
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Born Rule proof for freshmen

Although early quantum mechanics are taught in many freshman courses, the Born Rule is almost never proved at that stage. Is it even impossible to elementarily prove that the probability density is ...
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Many worlds interpretation and probabilities

How is the many worlds interpretation (MWI) of QM consistent with the probabilistic interpretation of the wave function (given by Born's interpretation)? For example, say a particle has a 90% chance ...
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Probability of a system being in a particular state after some time $T$

If we study a QM system and an observable that we can measure on it (like Energy etc). I know that for the probability that the system is in an eigenstate (discrete and not degenerated) we have ...
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I understand what it represents, but what physically is the wave function?

In quantum mechanics, I understand that the wave function represents the state of a particle and that the square of the wave function tells us the probability of a particle being found at a particular ...
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Origin of probabilities in Quantum Mechanics?

The non-normalized wavefunction of a general qubit is given by: $$|\psi\rangle=A|0\rangle+B|1\rangle.$$ The complex amplitudes $A$ and $B$ can be represented by two arrows in the complex plane: Now ...
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Correct interpretation of $\langle x | \psi \rangle$?

Suppose $|x\rangle$ is an eigenvector of the position operator $\hat{x}$ and let $|\psi\rangle$ be an arbitrary state on this Hilbert space. What is the correct interpretation of the complex number $\...
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Orthonormality condition in quantum mechanics [closed]

What does the orthonormality condition in quantum mechanics truly signify? Does it have a physical meaning? Or is it just a method of normalization applied in order to find the probabilities?
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Expectation value and the probability of finding a particle [closed]

I'm trying to understand basic quantum physics, as I understand, the expectation value of some random distribution gives us the outcome that we might expect(highest probability) if the event is done ...
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What is the meaning of probabilities in quantum mechanics?

In quantum mechanics, probabilities are associated with the detection of a physical event by a macroscopic device, or are events at the microscopic level also probabilistic? For example, the ...
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Why $\psi(x)$ needs to be square integrable?

The question might be due to some incompleteness. A state vector ket, $|\psi\rangle$ can be expanded in terms of position basis as, $$|\psi\rangle = \int |x\rangle \langle x|\psi\rangle dx$$ Now if I ...
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Measuring a superposition state of identical particles

Suppose that the state of a system of two identcial particles (say, two photons) is given by the following: $$\frac{1}{\sqrt{}2}(|\psi_{1}\rangle|\psi_{2}\rangle + |\psi_{2}\rangle|\psi_{1}\rangle) \, ...
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Are classical probability a special case of quantum probabilities? How to show it?

First, let's say I have a classical system involving throwing a fair coin. There are two possible events $\{\text{head},\text{tails}\}$. Their respective probabilities are: $$ P(\text{head})=\frac{1}{...
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What are we measuring in a quantum field when we square the wavefunction?

Suppose we are doing a measurement in a particular quantum field, i.e electron field. Are we looking for the probability of the electron to show up at that spot we are measuring or are we measuring ...
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Is it possible to create a consistent quantum theory such that the probability is the complex norm, instead of the square complex norm?

In quantum physics, the relation $$ \int_{-\infty}^{\infty} (\psi[x,t]^*)(\psi[x,t]) dx=1 \tag{1} $$ is paramount. What would the consequence be of defining the normalization condition as $$ \int_{-\...
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Probability of a quantum particle [duplicate]

Now recently I have started quantum mechanics and I understood the wavefunction but I don't understand why $|Ψ|^2$ gives the probability density of a quantum particle. Is there a reason or perhaps ...
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How do we expand Bayes theorem to account for probability amplitudes?

The Bayes theorem simply states: $$ P(B | A) P(A) = P(A | B) P(B) $$ I wonder if there is something that can be meaningfully said as generalization of this relationship when the probabilities in ...
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What is the probability of measuring $p$ in the momentum space?

I have a wave function $\Psi (x,t)$. According to the Max Born postulate, $\lvert\Psi (x,t)\rvert ^2$ is the probability density. This quantity specifies the probability, per length of the $x$ axis, ...
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Why do we describe probability amplitude rather than probability itself in quantum mechanics?

In the quantum mechanics, the dynamics of quantum system are described in terms of probability amplitude. However, we want to calculate the probability in the end which can be measured. Why don't we ...
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“Probabilities are the ghosts of quantum mechanical amplitudes”

I came across this quote today; [Quantum computers] process information using quantum mechanical amplitudes. And probabilities are sort of the ghosts of amplitudes after they have been degraded to ...
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Zero Solution in infinite square well?

Consider the well: $$V(x) = \begin{cases} \infty&\text{if }x<0 \\ 0&\text{if }x\in\left(0,L\right) \\ \infty&\text{if }x>L. \end{cases}$$ Solving the time independent ...
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What is exactly Max Born rule?

I have thought Max Born rule as one of the axiom of quantum mechanics that says norm square of wavefunction gives the probability density. But I also found written somewhere that the rule says that ...
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123 views

What does a Hilbert space state vector represent in Koopman–von Neumann theory?

I understand what a state vector is in quantum mechanics. I also understand that in KvN theory, both the quantum Hilbert space and the classical Hilbert space are the same (see the answer to this ...
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Wave Function is case of Space-Time curvature

I have some questions about wave functions and space-time curvature: For what I have understood, Dirac Equation incorporates special relativity in Schrodinger equation, but still there are no ways to ...
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The origin and current usage of the phrase “transition probability” in Wigner's sense

The definition of transition probability by Wigner is the following The square of the modulus of the unitary scalar product $(\psi,\varphi)$ of two normalized wave functions $\psi$ and $\...
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Quantum probabilities or frequencies? Derivation or a postulate

The Born rule $$p=|\langle\psi|\phi\rangle|^2$$ defines the quantum probability and answers the question what is the probability that the measurement of $\psi$ will produce the outcome $\phi$? This ...
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Reference request for a mathematical motivation for the Born rule

I was reading the popular science book The Hidden Reality by Brian Greene. My question is about a part in the notes at the end of the book. It is chapter 8, note 9. Brian Greene describes a ...
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The justification for the probability of definite energy states in quantum mechanics

In quantum mechanics, if the energy of a system is measured at some $t$ the probability of obtaining the energy eigenvalue $E_i$ is: $$\left| \int_{-\infty} ^{\infty} {\psi_i^* (x)\Psi(x,t)} dx^2 \...
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Do all square integrable functions represent a probability when volume integrated? [closed]

I have just begun with Quantum Physics. My question is: do all square integrable functions represent a probability when volume integrated? Or is it only the wavefunctions under Born's interpretation? ...
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What happens in an infinitely long potential step when $E<V$?

In the case of a potential step when $E<V$, the transmission coefficient of the particle is zero. However, there is also an exponentially decaying wavefunction of the particle in that classically ...
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A simple explanation of the Born rule (v.2)? [duplicate]

Please post further comments or answers to A simple explanation of the Born rule? The probability that an initial quantum state $|\psi_i\rangle$ evolves to become the final quantum state $|\psi_f\...
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A simple explanation of the Born rule?

The probability that an initial quantum state $|\psi_i\rangle$ becomes the final quantum state $|\psi_f\rangle$ is given by \begin{eqnarray} P(i \rightarrow f) &=& |\langle\psi_f|\psi_i\...
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What made Born interpret $|\psi|^2$ as a probability density?

What was Born reasoning when he introduced the rule that $|\psi|^2$ could be interpreted as a probability density?
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QM probability density function without Born's rule, invariant to wave-function phase

The QM probability density as a function of the wave function is given by Born's rule as a postulate. This leads to the probability density being invariant to the phase of the wave function. Suppose ...
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Probabilities in Quantum Mechanics: Measurement Outcomes or More?

In all treatments of quantum mechanics, the probabilistic nature of the theory enters via the Born rule for the statistical properties of the measurement outcomes of some observable. In short, this ...
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What is difference in Dirac Notation for probability and Probability Density in Quantum Mechanics? [closed]

The Dirac Notation for wave function $$\langle\psi|\psi\rangle= \int_{-\infty}^\infty \psi^{*}\psi \,dx $$ $$\text{Probability} = \int_{-\infty}^\infty \psi^{*}\psi \,dx $$ But most often it is ...
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Understanding wave function graph

I found this graph from the internet that interprets the graphical representation of wave function.I completely understand the wave function that is depicted by blue line but i really am confused ...
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How much can a wave function tell us?

We can not predict the future by getting the velocity and position of particles since it’s not possible to get both of these together due to the uncertainty principle. But, according to Hawking’s book ...
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Question about a point in Srednicki's QFT book

On page 6, Sredniciki says (taking into account the erratum), that the "simplest possibility is for Alice and Bob to agree on the value of the wave function at a particular space-time point". This ...
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Bohr's Correspondence Principle and the Born Rule

Bohr's correspondence principle and the Born rule are related right? The correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical ...
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In the pilot-wave theory, is the quantum potential moving electrons randomly inside atoms?

As we know, in the pilot-wave theory (Bohmian mechanics), particles are guided on certain trajectories by the wavefunction. Here (In Bohmian mechanics, do electrons move inside an atom?) I asked about ...
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What is Wave Function? [duplicate]

Well, what is the meaning of wave function? What does it represent? In Schrodinger's equation, we find the value of Ψ. But what is Ψ exactly? Max Born only gave an explanation of what $Ψ^2$ (the ...
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Does the electron act as point charge in scattering theory?

We know that an electron behaves like a point charge, and the probability density of its position is given by the Born rule. Now suppose that we shoot an electron toward an atom, so the electron gets ...
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Is the Born rule indeed wrong?

This is a question about the validity of a preprint, arXiv:quant-ph/0509089, which claims that the "Copenhagen Interpretation of QM is incorrect" (same title, authored by Guang-Liang Li and Victor O.K....
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Does the Central Limit Theorem hold for position measurements?

A friend asked me recently if the Central Limit Theorem holds for quantum systems: i.e., if the distribution of measurements (e.g., of position) for any wavefunction would prove approximately normal, ...
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209 views

How to understand the kernel as a transition amplitude?

Consider the time evolution operator $U(t_f, t_i)$ that controls the evolution of a wave function according to $|\psi(t_f \rangle = U(t_f, t_i) | \psi(t_i) \rangle$. As I understand it, the Born ...
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How to evaluate the probability when a particle is detected?

Everyone knows the standard probability interpretation of the quantum mechanics. For example, the wave function of some particle at some time $t$ is $\psi (x,t)$. Therefore, if the particle is ...
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Does the wavefunction probabilities have to sum to 1? [duplicate]

In quantum mechanics we are often told that $\int |\psi(x,t)|^2 dx^3 =1$. i.e. the probabilities have to sum to 1. And that this implies the time evolution operator is unitary. But can't we define ...
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Normalisation of quantum states: why?

We all learn that quantum states need to be normalized, as they are associated to probabilities which needs to sum up to one. However, I would like to know whether you have other valid reasons to ...