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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 ...
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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 ...
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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,...
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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 $C$. ...
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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 ...
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 $\Delta{X}^2 = <X^2> - <X&... 5answers 7k views Differences between probability density and expectation value of position The expression$\int | \Psi\left(x\right)|^2dx$gives the probability of finding a particle at a given position. If wave function gives the probabilities of positions, why do we calculate "... 5answers 1k views About the definition of expectation value in quantum mechanics In quantum mechanics, the expectation value of a observable$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 ... 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 ... 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 ... 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]\...
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 \, \...