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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Understanding operator product of three mixed states with three projector operators

I am recently studying the triangle scenario in the context of Bell nonlocality (for reference, see for instance this article). In it, we have three parties, commonly referred to as Alice, Bob and ...
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On the Measurement Problem

In the orthodox interpretation of quantum mechanics, the following three assumptions are made (please correct me if I am wrong): Every physical system is completely specified by a state $\lvert\psi\...
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Why is psi square a possibility? [duplicate]

Is psi square just an assumption? Or there is a physical reason why they defined like that? My procedure is: It is intuitive for me to think possibility is proportional to energy distribution. ...
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Ontologically speaking, does single-shot quantum interference occur in pairs of possibilities?

I've been perplexed by the semantics used in Science 329, 418-421(2010), where they state that according to Born’s rule and its square exponent, interference always occurs in pairs of possibilities. ...
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What would Born's rule look like if it were to the power three?

In matrix notation, the probability that some quantum state $\hat{\rho}$ is measured along some projector $\hat{\Pi}_k$ is \begin{equation} p_k = \mbox{Tr}\{\hat{\Pi}_k \hat{\rho}\}. \end{equation} I'...
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A question about Born's rule in the Many-worlds interpretation

Can somebody explain the idea of self-locating uncertainty when there's a binary choice of eigenstates for an entanglement? I understand the idea that because you are unsure of which branch you are in,...
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Does quantum mechanics need projective representations only due to the Born rule?

In quantum mechanics, physical states don't live in the Hilbert space, but rather on the equivalence class of rays on the Hilbert space. This is called a projective space. This is the reason why when ...
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Does the Many Worlds interpretation trivialize the Born rule?

I'm thinking that the Many worlds interpretation trivializes the Born rule. Suppose we take an electron's spin's state vector such that the probability of spin up is 0.6 and that of spin down is 0.4. ...
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Understanding the probability of measurement w.r.t. density matrix

I am told that the probability of measuring $\lambda$ is $$p_\lambda = Tr(\hat{P}_\lambda\hat{\rho}) = Tr(\hat{P}_\lambda\hat{\rho}\hat{P}_\lambda)$$ where $\hat{P}_\lambda = \sum_{n:\lambda_n = \...
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Why does quantization lead to observed quantites becoming probabilities?

I am reading the nice discussion on this MO thread on the idea behind Quantization mathematically. This answer is quite nice, and for my question, I take some quotes from it: "quantum ...
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Non-normalizable eigenfunctions

I was reading Shankar's Principles of Quantum Mechanics when on page 65, he starts talking about infinite spaces and operators in them. He introduces an operator $K$, which in the $x$ basis takes the ...
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Probability density and wavefunction

I am trying to understand the link between the wavefunction and probability density. I can understand the probability density $\rho$ would be some function of the wavefunction, but I am unable to ...
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Do we actually need negative probabilities in quantum mechanics?

I was reading this thread and I'm a bit confused. The answer says negative probabilities can account for destructive wave interference and the events cancelling out. But if events just cancel out, ...
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Must a wavefunction be normalisable for a pdf to exist?

My course notes say, for normalised wave functions $\psi(x,t)$, the function $$\rho(x,t)=|\psi(x,t)|^2=\overline{\psi(x,t)}\psi(x,t)$$ gives the $\color{red}{\text{probability density}}$ for the ...
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The probability density function $|Ψ|²$ [duplicate]

Max Born stated that $|Ψ|²$ is the probability density of a particle, given its wave function to be $Ψ$. But why is this? Where does this come from?
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How are Gleason's and Kochen-Specker's theorems related?

If, on the one hand, I were to paraphrase Gleason's theorem, it would loosely state that if one can assign a truth value $p_k$ to each basis vector $\vec{u}_k$ such that $\sum_k p_k = 1$, then that ...
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Why do we only look at transitions between energy-eigenstates, when perturbing a system with an oscillating interaction?

I have some doubts about the usual explanation of sharply peaked emission / absorption spectra, that one can observe when one looks at quantum mechanical systems, for example the hydrogen atom. In ...
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If states cannot be written as superposition of eigenkets of observable, then how do we measure an observable for that state?

Usually, if we have a state $|\psi\rangle$, and have to measure an observable $A$, then all we do is expand $|\psi\rangle$ in terms of the eigenvectors of observable A, and then the probability of ...
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Why it is a longstanding challenge to reproduce Born rule in Everettian QM?

I'm reading this Sebens and Carroll's paper on Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics. Where they presented their derivations of the Born rule and ...
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Nature of expectation values and Born's rule and the measurement problem

Suppose we take a normalised quantum mechanical wave function of $\Psi (\mathbf{r} ,t)$. If we expand it in a certain form of spatial functions $\psi_{n} (r)$ which is complete orthonormal. Then we ...
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Is the Born rule a red herring in explaining the measurement problem? [closed]

Many explanations of the measurement problem try to derive the Born rule from Schrodinger evolution, for example Many worlds. I have two reasons to think the Born rule isn't fundamentally related to ...
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Transition from discrete case to a continuous case with regards to the Born's Rule

I learned that given that the eigenvalue equation is $$ \widehat{A}\left|u_{n}^{i}\right\rangle=\lambda_{n}\left|u_{n}^{i}\right\rangle $$ where $ i \in\{1,2, \ldots, g_n\} $, and that the state $ |\...
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Probability of a state "being found in another" [closed]

I'm sorry if something gets lost in translation, as my professor wrote all questions in portuguese, but what does it mean to ask the probability of finding "state $|\alpha \rangle$ in state $|\...
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Replacement of Born rule to understand consciousness [closed]

This postulate should replace the Born rule as it makes the Born rule precise w.r.t. decoherence: "If a consciousness is in a superposition $\sum |k_i \rangle$, such that $\langle k_i|k_j\rangle=...
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Basic Quantum mechanics: This nonphysical "fun" homework with operators just gives me nonsense [closed]

For QM homework we are given $\vert \psi \rangle = \sum_{i = 1}^3 c_i \vert i \rangle$ where $\vert i \rangle$ represent different positions. Firstly we are asked $P(i=2\vert \psi)$ which is given $P(...
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Which experiments confirm that the Born rule is $|\psi|^2$ rather than $|\psi|$?

It seems like some experiments on quantum systems, like the electron $g-2$ measurement, do not rely directly on the Born rule, since they are more so measuring inherent characteristics of the ...
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I'm trying to understand this paper on Born rule in Many-Worlds Interpretation

https://arxiv.org/abs/1405.7907 In particular I would like to understand this paragraph, right before chapter 6: This route to the Born Rule has a simple physical interpretation. Take the wave ...
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Does Born's rule guarantee component-wise probabilities? [closed]

Suppose I have the wave function $\vec{\psi} = \begin{bmatrix} \psi_x \\ \psi_y \\ \psi_z \end{bmatrix}$. If I understand Born's rule correctly, the equality $\vec{\psi}^* \cdot \vec{\psi} = f(x,y,z)$ ...
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Born's Rule for states over supernumbers?

For Quantum-mechanics on a Hilbert-space over the complex numbers, the usual scalar product of two states $\langle \phi | \psi \rangle$ and gives the transition amplitude between the two states. The ...
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A question of the semantic meaning of the (non-relativistic) propagator

In the Wikipedia article it says "the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time". Let us ...
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Interpretation of Born Rule In QFT

Can we born rule be used to find probability of a particle to exist in a region in QFT using the formula $\int_a^b \psi(x)\psi^*(x)dx$,where $\psi(x)$ is a fermionic field? If yes, please provide ...
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Wavefunction Amplitude Intuition

Reading the responses to this question: Contradiction in my understanding of wavefunction in finite potential well it seems people are pretty confident that, e.g., the wavefunction of a particle in a ...
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Probabilities of eigenfunctions

I am struggling to understand how to get the probabilities of each eigenstate occurring from a wavefunction that is a linear combination of eigenfunctions. If we have a wavefunction $$\Psi = A ( e^{...
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What is an intuitive or simple proof of Gleason's theorem and how it relates to the Born rule?

What is an intuitive or simple proof of Gleason's theorem and how it relates to the Born rule? I tried to read the articles, but the proof seemed big and the kind that are unintuitive (im not ...
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How are can there be initial conditions in Bohmian mechanics that disobey the Born rule if Gleason's theorem is true?

Gleason's theorem constrains the possible measures that are allowed on Hilbert spaces of dimension $\ge 3$. It is often said that Gleason's theorem essentially says that the Born rule is the ...
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How is it possible for Born's interpretation of the wave function to be published after Schrödinger published his equation?

If I am right Born published his interpretation of the wave function after Schrodinger published his wave equation. However, according to my QM textbook, all the expected values of quantities (like ...
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Why macroscopic bodies should exist as wavepacket?

Based on my understanding, we assume that the electrons, exist as wavepackets in the solids while deriving the transport equations for transistors, we create wavepackets out of momentum eigenstates ...
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Why doesn't Gleason's theorem imply the Born rule?

I know that the question "does Born's rule follow from Gleason's theorem" has already answers on the website: see here, and here. I am not satisfied with the answers given (one cannot rule ...
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Why particularly probability density is defined as $|\Psi|^2=\Psi \Psi^{*}$?

It may be a stupid question, but why particularly for probability density expression $k~|\Psi|^2 = k~\Psi^{*}\Psi$, it's assumed that $k=1$? As it is now, then in a complex plane probability density ...
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What is the relation between the wave function in Born's rule and the wave function in Dirac's equation? [duplicate]

The wave function for spins without position can be seen as a complex wave vector $\psi=(\psi_1,\psi_2,\ldots)$ and the probability to measure a state $\psi^{(A)}$ in another state $\psi^{(B)}$ is $$ ...
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What is the distribution for a function of different quantum observables?

Suppose we have a quantum mechanical particle prepared in a pure state $\psi$, and an apparatus that can measure the orbital angular momentum of the particle along a specified orthogonal axis ($x$, $y$...
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Can I interpret the squaring of the wave function like this? [duplicate]

Born rule states that the probability density of the wave function is equal to the square of the function over the given interval. I thought, "Why squared?". I came up with this: "We ...
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Interacting Picture in QFT

I'm having trouble understanding how the interaction picture describes scattering. In quantum theory, the probability amplitude for a system in state $|i(t_i) \rangle$ to be measured in state $|f(t_f) ...
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How to derive Collinear amplitude proportional to Born amplitude

In the collinear limit, the squared matrix element factorises into (for partons 4 and 5 going collinear) \begin{eqnarray} \overline{\sum}|M_3(1+2 \to 3+4+5)|^2 \approx \overline{\sum}|M_2(1+2 \to 3+4')...
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Probabilities fall outside the differential?

A few years ago I had a conversation with a physical chemist, and one of her comments still lingers in my memory. Our conversation was about Quantum Chemistry. In response to my commenting on the ...
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Born rule for a sequence of measurements: Why this particular form?

If we have some observable $O=\sum_i\lambda_iP_i$ where $P_i$ are the usual projectors you get in a spectral decomposition, then the probability for a single measurement yielding an outcome $\lambda_j$...
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Is the Born rule usually regarded an axiom in quantum mechanics?

The statement For simplicity, let's consider a finite-dimensional Hilbert space. (The question can probably be generalized, but I don't know enough about mathematical QM to properly do so.) Let $A\...
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What does Hartle's derivation of the Born rule actually amount to?

There have been many questions asked here on the topic of whether the Born rule can be derived from the rest of the axioms of quantum mechanics. See, for example, this and links therein. However, I ...
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Probability current (Integral in all space)

So , when we take the integral in all space of the probability current j (as defined in the first relationship here https://en.wikipedia.org/wiki/Probability_current) in non relativistic quantum ...
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Wave function and uncertainty principle

Why is the position of a particle in the Schrödinger wave equation represented as an exponential periodic wave $$A\exp\left(\frac{(2\pi\iota)(px-Et)}{h}\right)$$ where $p$ is momentum and $E$ is the ...
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