[2211.12742] Spectral theorem for dummies: A pedagogical discussion on quantum probability and random variable theory # Spectral theorem for dummies: A pedagogical discussion on quantum probability and random variable theory Andrea Aiello andrea.aiello@mpl.mpg.de Max Planck Institute for the Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany ###### Abstract John von Neumann’s spectral theorem for self-adjoint operators is a cornerstone of quantum mechanics. Among other things, it also provides a connection between expectation values of self-adjoint operators and expected values of real-valued random variables. This paper presents a plain-spoken formulation of this theorem in terms of Dirac’s bra and ket notation, which is based on physical intuition and provides techniques that are important for performing actual calculations. The goal is to engage students in a constructive discussion about similarities and differences in the use of random variables in classical and quantum mechanics. Special emphasis is given on operators that are simple functions of noncommuting self-adjoint operators. The presentation is self-contained and includes detailed calculations for the most

# Why is a Hermitian operator a "quantum random variable"? Tags: quantum-mechanics, mathematical-physics, operators, probability, observables - Score: 24 - Views: 3259 - Answers: 7 - Answered: yes - Asked by: user79317 - Asked: 2016-06-14 - Edited: 2016-06-14 - Site: physics ## Question 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, and probabilities of these events. A random variable is just assigning these numbers. I took QM as an undergrad and I remember computing eigenvalues, expectations, etc. of various operators and I never quite got what an operator is in QM. A (quantum) random variable is a Hermitian operator on some Hilbert space. You can compute probabilities and expectations by some formulas involving eigenvectors and orthogonal projections. I must admit, even after my QM class, I don't get this. Is this random variable in any way related to my ignorant definition? Could we model say a coin flip or dice roll using this? Or is this type of random variable only for quantum things? Why not model all of quantum mech

Observable In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is described by a linear operator. For example, these operators might represent submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question. ## Quantum mechanics Every observable quantity in a quantum system is represented by a linear operator. John Archibald Wheeler used the analogy of a machine to describe operators: a quantum state goes in to the machine and the result state comes out. The result state will be one of the eigenstates of the operator. If the input was an eigenstate, the output will also be that eigenstate. In all other cases the output

# Why do we use Hermitian operators in QM? Tags: quantum-mechanics, operators, hilbert-space, eigenvalue, observables - Score: 35 - Views: 20903 - Answers: 4 - Answered: yes - Asked by: Benjamin Hodgson (1064 rep) - Asked: 2012-10-11 - Edited: 2016-04-28 - Site: physics ## Question Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') operators. The eigenvalues of the operator are the allowed values of the observable. Since Hermitian operators have a real spectrum, all is well. However, there are non-Hermitian operators with real eigenvalues, too. Consider the real triangular matrix: $$ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 8 & 4 & 0 \\ 5 & 9 & 3 \end{array} \right) $$ Obviously this matrix isn't Hermitian, but it does have real eigenvalues, as can be easily verified. Why can't this matrix represent an observable in QM? What other properties do Hermitian matrices have, which (for example) triangular matrices lack, that makes them desirable for this purpose? ## Answers ### Answer by Qmechanic (score: 40 [ACCEPTED]) One problem