Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

3
votes
1answer
82 views

Two Time Correlation function calculated from Born rule

Update Below I'm having a hard time reconciling two different calculations of the quantum two time correlation function. Consider quantum operator $A$ with eigenvectors $\{|\phi_i\rangle\}$ and ...
0
votes
0answers
49 views

Generalizing Born rule

In Quantum mechanics, The born rule is that the probability is given by $\mid \langle\Phi\mid\Psi\rangle\mid^2$ I understand that this is essential in order that the QM reduces to the correct ...
1
vote
1answer
54 views

A textbook's reading of the Born Interpretation

A textbook states the following: $P(x, t) \ dx $ is the probability that a measurement of the position of the particle described by $\Psi (x, t)$, at time $ t$, will find it in the region $(x, x +...
1
vote
2answers
68 views

I've read that $\langle a | b\rangle$ is a probability amplitude but $\langle a | a\rangle$ is a probability. Why the inconsistency?

I'm studying elementary quantum mechanics, and I've read that $\langle a \vert b \rangle$ is the probability amplitude of a transition from state $a$ to state $b$. Thus, $|\langle a | b \rangle|^2$ ...
-2
votes
1answer
72 views

Does a wave function never reach zero probability density?

$$ψ=e^{iκx}$$ Since the wave function is an exponential equation, is there no point with zero probability density of finding a given particle? Does that justify quantum tunneling?
0
votes
2answers
76 views

Finding amplitude of probability

The mathematical structure of quantum mechanics, follows almost inevitably from the concept of a probability amplitude. For James Binney and David Skinner: "With every value in the spectrum of a ...
0
votes
1answer
78 views

What is the physical basis of Born's interpretations?

Did anyone has any idea how Born came up with the probabilistic interpretation of quantum mechanics. It is by all means very bizarre. And then it leads to the idea of copenhagen interpretation. Also ...
0
votes
1answer
144 views

Probability of finding an energy state of a non-normalisable wave-function

Suppose, say, I have the following wave function It represents the wave function of a free particle. I would want to calculate the probability of finding the particle with energy ħk and energy 2ħk. ...
12
votes
2answers
298 views

Probability measure implies quantum mechanics?

The article "Quantum Logic and Probability Theory," by Wilce, has the following in section 1.4: 1.4 The Reconstruction of QM From the single premise that the “experimental propositions” ...
9
votes
3answers
668 views

Does Gleason's Theorem Imply Born's Rule?

Suppose that I accept that there is wave function collapse in quantum mechanics, and that the probabilities associated with each orthogonal subspace are a function of the wave function $\psi$ before ...
2
votes
0answers
58 views

Reconstruct quantum state from probabilities

Given a quantum state, the Born rule lets us compute probabilities. Conversely, given probabilities, can we reconstruct the quantum state? I think the answer is almost trivially positive but how ...
0
votes
1answer
112 views

Born rule and expectation value integral for $L = x \times p$

Physical interpretation of the wave function $\psi (x,t)$ is a probability amplitude for location $x$ and the Fourier transform of $\psi (x,t)$ can be interpreted as a probability amplitude for ...
-2
votes
4answers
101 views

Can the complex square of the wave function be interpreted as shape of the particle? [closed]

We know that for wave function of a photon or an electron, the complex square of the wave function is understood as the probability density of finding point like particle in the location. Can anyone ...
2
votes
0answers
391 views

Validity of Born Rule [closed]

The Born rule has been very successful in quantum mechanics. However, the interesting fact about this rule is that it only allows pairwise interference. In other words, there are no interference terms ...
-1
votes
1answer
108 views

Does the Born rule imply $L_2$ Space?

I see no formal proof of the Born rule. Well, the normalizing condition $\int_\infty|\Psi|^2dx=1$ is because of Born rule if I am not wrong. Does this imply that our reality is a $L_2$ space? If ...
5
votes
1answer
129 views

Implication of Born's rule on the superposition principle

BACKGROUND Born's rule quantifies the interference pattern of a single quantum particle going through two possibles paths A and B as $P = |A|^2 + |B|^2 + ⟨A|B⟩ + ⟨B|A⟩$. The standard interpretation ...
0
votes
1answer
309 views

If the Many Worlds interpretation (MWI) cannot derive the Born rule, would that mean that it is wrong?

All attempts at deriving the born rule in MWI have been shown to be circular in some way. So if it turns out that MWI cannot derive the born rule without some form of circularity, does that mean that ...
2
votes
1answer
198 views

Are there any derivations of the born rule from MWI that work?

Are there any derivations of the born rule from MWI that are not circular and are mathematically consistent? I ask this because Florin Moldoveanu insists that such a derivation from MWI doesn't exist. ...
0
votes
2answers
109 views

What is the initial and rigorous definition of state function $\psi$ in quantum mechanics?

Is the $$ P_{a\le x\le b} (t) = \int\limits_a^b d x\,|\psi(x)|^2 \, $$ and $$ P_{a\le p\le b} (t) = \int\limits_a^b d p\,|\psi(p)|^2 \, $$ results or axiomatic definitions? Sorry if the ...
3
votes
2answers
343 views

What precisely is the wave function a probability density of?

In QM the norm of the wave function $\psi(\vec{x})$ is said to be the probability density that the particle is at $\psi(\vec{x})$ if one would observe its position. Generally, nothing more is ...
0
votes
1answer
87 views

Defining Quantum Mechanics

Does Schrödinger's Equation (Operator form) $[\hat{X},\hat{P}]=i$ Born Rules define Quantum Mechanics?
0
votes
1answer
77 views

Understanding the Mathematical Formalisms of Everetts MWI

Link to article: http://jamesowenweatherall.com/SCPPRG/EverettHugh1957PhDThesis_BarrettComments.pdf I'm writing an essay on Everettian MWI and its incompatibility with Born Rule probabilities. I ...
0
votes
1answer
52 views

Basic question about wave funtions: using the Born rule

I have a textbook giving an example of what the probability is of observing system $Ψ = a|A⟩ + b|B⟩$ in states $a|A⟩$ and $b|B⟩$. I'm not sure I understand it fully. How do I use the Born rule to know ...
1
vote
2answers
146 views

Clashing definition of rays in Weinberg and Sakurai and Born interpretation without normalizability [duplicate]

In Sakurai's Modern Quantum Mechanics, it is stated that One of the physics postulates is that $|\alpha\rangle$ and $c|\alpha\rangle$, with $c\neq 0$, represent the same physical state. In other ...
2
votes
1answer
101 views

Algebra behind the wave function properties [closed]

In lecture, for the tunnelling wave function $$ \psi(x) = C_1\cosh(x/l)+C_2\sinh(x/l)$$ the current density is $$ J = h/(2mi) [ \psi^*(\Delta\psi) - (\Delta\psi)^*\psi] $$ Here is my problem, ...
0
votes
2answers
2k views

Physical interpretation of nodes in quantum mechanics

In Schrödinger's approach to quantum mechanics, we talk about the probability of finding a particle in a definite location in space. Now if we look at a simple quantum mechanical system, say the ...
1
vote
1answer
170 views

Is the wave function the Radon-Nikodym derivative of a complex measure?

I read somewhere (latest version of a webcomic to be honest) that "superposition" means: a complex linear combination of a $0$ state and a $1$ state... Quantum mechanics is just a certain ...
0
votes
3answers
218 views

Born's interpretation of psi

According to the Born's postulates, psi should be atleast differentiable to the first order then why does we not require psi in this case to be differentiable at X = a and X = 0.Also psi comes out to ...
3
votes
1answer
573 views

Why to take the absolute value of the wave function before squaring it?

For obtaining the probability distribution we should take the absolute value of the Schrodingër Wave Function 'Ψ'and then square it. But why to take first the absolute value if the square is going to ...
0
votes
2answers
125 views

Probability of quantum transition

I have a question about a task: We have a particle, which is in a linear combination of the first two states of the harmonic oscillator, which we can parametrise as $|\psi\rangle=\cos(\frac{\...
3
votes
1answer
113 views

What are some experimental verifications of Born's rule in quantum mechanics?

Born's rule in quantum mechanics states that when measuring a system using a measuring device that can detect (=project onto) an orthogonal basis of states, the probability of obtaining a certain ...
0
votes
3answers
134 views

How does technology rely on the probabilistic nature of quantum mechanics?

With respect to the dual slit experiment and the conclusion of probability waves, I was watching a documentary that said without "accepting chance" we couldn't have functioning technology in today's ...
6
votes
2answers
2k views

What is radial probability density?

I have read the pages suggested as similar and not found the answer to my question, or if it's there I didn't understand it. I have seen radial probability density described on various different ...
1
vote
0answers
91 views

Consistency of “Adding Amplitudes” and Normalization

Consider two particles ($ a $ and $ b $) colliding and scattering elastically at right angles (calling paths 1, 2; up and down, respectively). The problem is to work out the probabilities of this ...
1
vote
0answers
47 views

Validity of analysing spherical harmonics in real-space using the probability amplitude [closed]

While looking up spherical harmonics (on the validity of analysing them in real-space in a transition metal crystal structure), I came across this: http://shpenkov.janmax.com/hybridizationshpenkov.pdf ...
0
votes
0answers
245 views

Why do we square probability amplitude? Why not direct values?

According to my knowledge (which is feeble) we will also get the same result if used direct values. For example, if the probability of something happening is 4% or 0.04, we should make an arrow of 0....
1
vote
2answers
96 views

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 ...
1
vote
1answer
149 views

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 ...
3
votes
1answer
610 views

Can a particle pass through a nodal point where its wave function is zero?

Let's consider an infinite square well. In the first exited state there is a node at the middle of the well (i.e. wave function and thus probability of finding the particle is zero there). If I ...
3
votes
0answers
523 views

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....
1
vote
1answer
107 views

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 ...
3
votes
2answers
150 views

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 ...
-3
votes
2answers
149 views

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 ...
0
votes
1answer
61 views

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\...
24
votes
4answers
29k views

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 simple ...
0
votes
1answer
72 views

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?
11
votes
4answers
533 views

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 ...
0
votes
2answers
376 views

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 ...
2
votes
0answers
86 views

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}^{\...
1
vote
1answer
267 views

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, ...