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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 \, \...