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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Born's interpretation for momentum operator

Hi I have a basic QM question: Given a state vector $|\psi(t) \rangle$, at some time $t$, we can project this onto the position basis, $\langle \vec{r}| \psi(t) \rangle = \psi(\vec{r},t)$. Then from ...
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Probability in QM: derivation or interpretation? [duplicate]

It is known that coordinates $C_k\in\mathbb{C}$ of the QM-state vectors $|\psi\rangle$ has an interpretation as probability weights $p_k$ in the whole state through the formula like $|C_k|^2=p_k$. We ...
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Schrödinger's interpretation of his wave function before Born

The below shows some excerpt from Feynman's lecture notes. 21–4 The meaning of the wave function When Schrödinger first discovered his equation he discovered the conservation law of Eq. (21....
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Different postulates and statistical interpreations of quantum mechanics

Hi I have a query about the difference of two aspects of the statistical interpretation of quantum mechanics given in the popular introductory quantum mechanics books "Introduction to Quantum ...
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What are “interferences of higher order” in the context of Born rule and triple-slit diffraction?

This question relates to the paper commented in this 2010 article. The paper itself is Ruling Out Multi-Order Interference in Quantum Mechanics; it is the discussion of a triple-slit interference ...
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What makes the probability distribution of a wavefunction in QM intrinsic? [closed]

I know that the usual interpretation of the wavefunction in QM is that it´s associated with a probability distribution of measurable quantities. Not a deterministic probability (like the probabilities ...
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Multiplication of associated probabilities

If a state $\psi $ is in the $ S_{z} $ basis represented by $\mid\psi\rangle = c_{+}\mid z\rangle + c_{-} \mid -z\rangle$ Does the associated probabilities change when I multiply $ \psi $ by $ e^{i\...
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What is a wave function in simple language?

In my textbook it is given that 'The wave function describes the position and state of the electron and its square gives the probability density of electrons.' Can someone give me a very ...
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How to calculate probability of complex wave functions? [closed]

An election has an equation as such: $$Ψ(x) = e^{iαx^2}.$$ How am I supposed to find the probability of finding the electron over a certain range? Is Fourier Transform involved in this?
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Connection between Hamiltonian version of the least action principle and probability amplitude in the Schrödinger equation

If I'm not mistaken, Schrödinger was influenced to look at wave equations because of de Broglie's assertion about particles having a wavelength. He started with the Hamiltonian equation which is ...
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Interpretation of position projection operator

Does the operator: $$\int_a^b \mathrm{d}x \, |x\rangle \langle x|$$ have physical meaning acting in a Hilbert space where it does not exactly correspond to the identity operator? Does it correspond ...
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Does an expectation value over only part of a wavefunction have physical meaning?

Does the expression: $$\langle p\rangle_{x=a, b} =\frac{\int_a^b \Psi(x)^*\,\hat{p}\,\Psi(x)dx}{\int_a^b |\Psi(x)|^2dx} $$ have any physical meaning when $\int_a^b |\Psi(x)|^2dx\neq\int_{-\infty}^{\...
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Expectation Value of Unitary Time Evolution Operator in Quantum Mechanics

Does the expression $\langle \Psi_i|U(t)|\Psi_i\rangle$ have a specific meaning, where $U(T)$ is the unitary time evolution operator of $\Psi$, and $\Psi_i$ is the initial state of $\Psi$? If so, ...
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Why does the preservation of transition probabilities imply the preservation of all quantum probabilities?

I have a question about symmetries in quantum mechanics. Let $H$ be a Hilbert space, and $\mathbb{P}H$ the corresponding projective Hilbert (ray) space. In quantum mechanics, a symmetry is usually ...
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Confusion about quantum probabilities depending on how coarsely grained the measurement apparatus is

I'm struggling to understand a confusing feature of the Born rule in quantum mechanics, which is that the probability of an outcome appears to depend on how coarsely grained our measurements are. ...
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Physical interpretation of the constant coefficient appearing in solution to the Schrodinger equation

The product solution to the Schrodinger's equation is $$\Psi_{n} \left ( x,t \right )=\psi\left ( x \right )\phi\left ( t \right )$$ By superposition, the solution becomes $$\Psi \left ( x,t \...
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How is the Born rule consistent with unitary evolution?

Consider a system $|\Psi_T \rangle_{t = 0} = |\Psi_E \rangle \otimes |\Psi_S \rangle$ where $|\Psi_S \rangle$ is a system that collapses into an eigenstate upon measurement. $|\Psi_E \rangle$ is the ...
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Measure-theoretic maths behind Born's probabilistic interpretation of Schrodinger's equation

I was reading a bit about Quantum Mechanics, Schrodinger's equation and its probabilistic interpretation (found this very insightful intro here https://plus.maths.org/content/schrodinger-1), my ...
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Is there a mathematical basis for Born rule?

Wave function determines complex amplitudes to possible measurement outcomes. The Born Rule states that the probability of obtaining some measurement outcome is equal to the square of the ...
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Generalized Born Rule for partial measurements

I'm looking for the mathematical formulation of this trait (from an introductory Quantum Computing course): How can I generalize this into higher dimensions? Or other observables? It does not seem ...
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Wave Function concept

What do we mean when we say wave function of electron? Does it mean wave nature of electrons? I am really confused.Without clearing this confusion i cannot proceed to molecular orbital theory.I am ...
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Can the Born rule be derived? [duplicate]

$\renewcommand{ket}[1]{|#1\rangle}$ If we have a particle and we know the initial state $|\psi\rangle$ of everything that is relevant, and we know the full Hamiltonian $H$, then we should be able to ...
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Born Interpretation of Wave Function

I have just started Griffiths Intro to QM. I was studying Born's interpretation of Wave function and it says that the square of the modulus of the wave function is a measure of the probability of ...
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Possible values of an observable are the eigenvalues of an operator

Ok, so I'm beginning to study quantum mechanics. For reference, the book I'm using is "Konishi-Paffuti/Quantum Mechanics-A New introduction". Now, I get that the quantum state of something (say, a ...
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Does Quantum mechanics predict (statistical) frequencies?

I began reading a book by Bricmont and Zwirn (Philosophie de la mécanique quantique, as yet not translated). In a note (page 4), Bricmont writes (translation mine): Probability is a theoretical ...
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Why does the magnitude squared of the wave function give us the probability density? [duplicate]

My question doesn't go much beyond the title: Why does $$\left | \psi \left ( x,t \right ) \right |^{2}$$ give us the probability density of something appearing at a certain location? I understand ...
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Expansion of a ket-physical interpretation of coefficients

Consider I have a state represented by the Ket: $$|\psi\rangle=\sum_i a_i |\phi_i\rangle$$ What are the physical interpretations of the coefficients $a_i$? My guess is that $|a_k|^2$ represents the ...
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Quantum Physics - What is the probability of it being in specific state (Stuck on question) [closed]

The normalised wavefunction for an electron in an infinite 1D potential well of length 65 pm can be written: $$\psi=(0.038 \psi_{n=1})+(-0.227\ i \psi_{n=10})+(g \psi_{n=5}).$$ If the state is ...
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What is the meaning of “ Ψ is not a measurable quantity in itself”?

I want to know that why the wavefunction Ψ as a complex quantity (i.e $A+iB$ form) in quantum mechanics and somewhere I have studied that Ψ is not a measurable quantity in itself that's why we ...
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How to use the Born rule to find the expected outcome of this simple Stern-Gerlach experiment [closed]

The experiment is shown below. How do I calculate the probability of observing a count in detector A, B, or C? Sakurai's text for example starts out describing how to calculate the outcome of simpler ...
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Normalization of wave function meaning…?

I just have one question. I'm doing a problem where I'm told to normalize a wave function, which is split up into two regions, namely where $r \leq r_0$ and $r > r_0$. My question is, why am I ...
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Does $\lvert\langle p\lvert\psi\rangle\rvert^2$ have any meaning at all?

I used to think $\lvert\langle p\lvert\psi\rangle\rvert^2$ had the meaning of some likelihood of the particle's momentum being $p$ (within some tolerance interval $\Delta p$). Now I'm just confused. ...
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Born-like measuring rule in classical experiments

this 2011 paper "Born's rule from measurements of classical signals by threshold detectors which are properly calibrated" by Khrennikov investigates the theoretical possibility of Born-like ...
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QFT propagator, time reversal and the Born rule

As far as I understand it a propagator, $D(x-y)$, gives the amplitude for a flow of positive energy-momentum from an earlier event $y$ to a later event $x$. Addendum: Instead of talking about energy ...
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Question about interpreting probabilities in QM [duplicate]

For the example of an infinite square well, $\psi(x)=0$ for $x$ outside the well/interval, and we are to interpret this as the particle cannot be found outside the well because $|\Psi(x,t)\bar{\Psi}(x,...
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Born's rule and Schrödinger's equation

In non-relativistic quantum mechanics, the equation of evolution of the quantum state is given by Schrödinger's equation and measurement of a state of particle is itself a physical process. Thus, ...
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Was Max Born the first to notice a connection between quantum mechanics and randomness?

Max Born introduced the Born Rule in a paper from 1926. But was this really the first time that a connection between quantum mechanics and randomness was noticed? Today, quantum mechanics and ...
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Are there two aspects of Born's rule?

I am having some problem understanding Born's rule. I am getting a little bit confused. Here it goes; Let $f(x,t)$ be a solution of Schrodinger equation. Then Born's rule says that the square modulus ...
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Can you get Born probabilities that are irrational from an experiment?

Is it possible to design an experiment getting irrational number predictions (in practice, you can't actually measure infinite numbers of decimal points, obviously) for Born probabilities? If you ...
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Decoherence in Everett quantum mechanics

Take an initial state and its environment, $E$, as follows, $$ |\psi\rangle_i = |0\rangle |E\rangle + \sqrt{2}|1\rangle |E\rangle. $$ Suppose that I've written it already in the basis in which the ...
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Is there any operator behind probability, in quantum mechanics?

In Quantum mechanics, the probability of finding a particle at position $x$ is given by $|\psi(x)|^2$, where $\psi$ is the wave function. Wonder what is the operator which gives this probability? Is ...
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Why is $|\Psi|^2$ the probability density?

I am starting with Quantum Mechanics, learning online. I can't seem to find the reason for $|\Psi|^2$ being the probability density of finding an electron. They've just taken it for granted everywhere....
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What are the problems in trying to interpret the Klein-Gordon equation as a single particle equation?

What is the problem if we try to interpret KG equation as a single particle equation? Also I wish to know whether the born interpretation of wavefunction is applicable in relativistic quantum ...
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Interpretation: probability form probability amplitude (free particle)

If you compute the probability amplitude of a free 1D non-relativistic particle with mass $m$, located at position $x_0$ at time $t_0$, for beeing detected at some other point $x_N$ at time $t_N$ you ...
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Probability in Quantum Mechanics: General

How do I find the most probable value of position of a (non-Gaussian) wave function? Is it the same value as the expectation value of the position? And is it true that the most probable value of ...
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Why is classical physics not valid for a harmonic oscillator in its lowest energy state? [closed]

I am reading Born's interpretation of wave function in quantum physics by Eisberg & Resnick and I am not able to understand this description about comparison between the classical and quantum ...
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Real Part of the Wave Function

In Quantum Mechanics the square of the wave function is compared to a probability density. Is there no similar relation to waves in the sense that something meaningful can be ascribed to the real part ...
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How to understand wavefunction in quantum mechanics in math

I am reading some introduction on quantum mechanics. I don't understand all but I get the point that the wavefunction tells some probability aspects. In one book, they show one example of the ...
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The probability of finding the electron in the H-atom

In the book Arthur Beiser - Concepts of modern physics [page 213] author separates the variables in the polar Schrödinger equation assuming: $$\psi_{nlm}=R(r)\Phi(\phi)\Theta(\theta)$$ then there a ...
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Born's Rule, What is the Reason? [duplicate]

As far as I've read online, there isn't a good explanation for the Born Rule. Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives ...