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0
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3answers
76 views

Measurement of a State Not in the Eigenbasis of the Operator

Suppose I have a two dimensional Hilbert space $\{ |0 \rangle,|1\rangle \}$ with these states being orthonormal. Now suppose I have the Hamiltonian $H=|1\rangle \langle 0|+|0\rangle \langle 1| .$ It ...
0
votes
2answers
81 views

Understanding the calculation of expectation value

The expectation value (in sense of discrete probability) can be thought of as $$ \left<a\right>=\frac{1}{N}\sum\limits^{N}{Â }\psi $$ where $N$ is the number of experiments. As the number of ...
0
votes
3answers
747 views

Probability of measuring eigenvalue of non-normalised eigenstate

This came up while working on a question about measuring the angular momentum of a particle in a superposition of angular momentum eigenstates: Given that: $$\langle\theta,\phi|\psi\rangle \propto \...
1
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1answer
165 views

Defining Kraus operators with normal distribution

I am interested in defining Kraus operators which allow you to define quantum measurements peaked at some basis state. To this end I am considering the Normal Distribution. Consider a finite set of ...
0
votes
1answer
69 views

Quantum measurement operators defined with Poisson distribution

I have defined a set of quantum measurement operators of the form $$A_C = \sum _M \sqrt{Pr(M|C)} |M \rangle \langle M|$$ where $Pr(M|C)$ is the Poisson distribution $$Pr(M|C) = \frac{e^{-C}C^M}{M!}~~~~...
19
votes
7answers
2k views

Why is a Hermitian operator a “quantum random variable”?

To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
2
votes
1answer
164 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
1
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0answers
111 views

Probability flux

I was reading a text on Quantum Mechanics in which it said that $$\int{d^3 x \, j(x,t)} = \frac{\langle p\rangle}{m},$$ where $\langle p\rangle$ is the expectation value of the momentum operator at ...
-1
votes
2answers
416 views

Expectation value expression Quantum Mechanics

Whilst working on a project I kept stumbeling across two different expressions for the standard deviation $\Delta{X}^2 = <(X - <X>)^2 >$ and the other $\Delta{X}^2 = <X^2> - <X&...
7
votes
5answers
7k views

Differences between probability density and expectation value of position

The expression $\int | \Psi\left(x\right)|^2dx$ gives the probability of finding a particle at a given position. If wave function gives the probabilities of positions, why do we calculate "...
3
votes
5answers
1k views

About the definition of expectation value in quantum mechanics

In quantum mechanics, the expectation value of a observable $A$ is defined as $$\int\Psi^*\hat A\Psi$$ But in probability theory the expectation is a property of a random variable, with respect to a ...
1
vote
1answer
62 views

Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption?

In Dirac's The Principle of Quantum Mechanics he makes the general assumption that "if the measurement of the observable $\xi$ for the system in the state corresponding to $|x\rangle$ is made a large ...
3
votes
5answers
1k views

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 ...
2
votes
1answer
114 views

Statistical sum of physical quantities in a quantum system

Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true? $\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$ $\...
1
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3answers
1k views

Operators explaination and momentum operator in QM

I know and understand why equation below holds. But i am new to operator thing in QM and would need some explaination on this. $$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, \...