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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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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Probability resulting from uncertainty when the measuring device exactly clicks?

Background Let's say I have $2$ set of eigenkets of observables of a system $|x_i \rangle $ and $|p_j \rangle$ (which do not commute). Let's say I have a non-ideal detector in the sense the ...
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On Born's rule and Cauchy-Schwarz inequality

Citing Born's rule: If an observable corresponding to a self-adjoint operator ${\textstyle A}$ with discrete spectrum is measured in a system with normalized wave function ${\textstyle |\psi \rangle }...
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Why is the probability that one state $|i\rangle$ ends up in the state $|f\rangle$ given by $|\langle i|f\rangle|^2$? [duplicate]

I've come across this relation numerous times, textbooks use it as if it is obvious. But I have never come across a proof or an intuitive explanation about why is it true. It would be helpful if ...
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Can I apply the Born rule to a Dirac spinor?

How does a Dirac spinor such as: $$ \psi = \pmatrix{a_0+ib_0\\a_1+ib_1\\a_2+ib_2\\a_3+ib_3} $$ Connect to a probability? Can one apply the Born rule of this object?
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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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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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