Appendix: Memento of Quantum Mechanics
In the following pages we recall the basic definitions, notations and results of quantum mechanics.
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In the following pages we recall the basic definitions, notations and results of quantum mechanics.
31.1
Principles
31.1.1 Hilbert Space The first step in treating a quantum physical problem consists in identifying the appropriate Hilbert space to describe the system. A Hilbert space is a complex vector space, with a Hermitian scalar product. The vectors of the space are called kets and are noted |ψ. The scalar product of the ket |ψ1 and the ket |ψ2 is noted ψ2 |ψ1 . It is linear in |ψ1 and antilinear in |ψ2 and one has: ψ1 |ψ2 = (ψ2 |ψ1 )∗ .
31.1.2 Definition of the State of a System; Pure Case The state of a physical system is completely defined at any time t by a vector of the Hilbert space, normalized to 1, noted |ψ(t). Owing to the superposition principle, if |ψ1 and |ψ2 are two possible states of a given physical system, any linear combination |ψ ∝ c1 |ψ1 + c2 |ψ2 , where c1 and c2 are complex numbers, is a possible state of the system. These coefficients must be chosen such that ψ|ψ = 1.
31.1.3 Measurement To a given physical quantity A one associates a self-adjoint (or Hermitian) operator Aˆ acting in the Hilbert space. In a measurement of the quantity A, the only possible ˆ results are the eigenvalues aα of A.
© Springer Nature Switzerland AG 2019 J.-L. Basdevant, J. Dalibard, The Quantum Mechanics Solver, https://doi.org/10.1007/978-3-030-13724-3_31
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31 Memento of Quantum Mechanics
Consider a system in a state |ψ. The probability P(aα ) to find the result aα in a measurement of A is 2 P(aα ) = Pˆα |ψ ,
(31.1)
where Pˆα is the projector on the eigensubspace Eα associated to the eigenvalue aα . After a measurement of Aˆ which has given the result aα , the state of the system is proportional to Pˆα |ψ (wave packet projection or reduction). A single measurement gives information on the state of the system after the measurement has been performed. The information acquired on the state before the measurement is very “poor”, i.e. if the measurement gave the result aα , one can only infer that the state |ψ was not in the subspace orthogonal to Eα . In order to acquire accurate information on the state before measurement, one must use N independent systems, all of which are prepared in the same state |ψ (with N 1) If we perform N1 measurements of Aˆ 1 (eigenvalues {a1,α }), N2 p measurements of Aˆ 2 (eigenvalues {a2,α }), and so on (with i=1 Ni = N), we can determine the probability distribution of the ai,α , and therefore the Pˆi,α |ψ 2 . If the p operators Aˆ i are well chosen, this determines unambiguously the initial state |ψ.
31.1.4 Evolution When the system is not being measured, the evolution of its state vector is given by the Schrödinger equation ih¯
d |ψ = Hˆ (t) |ψ(t), dt
(31.2)
where the hermitian operator Hˆ (t) is the Hamiltonian, or energy observable, of the system at time t. If we consider an isolated system, whose Hamiltonian is time-independent, the energy eigenstates of the Hamiltonian |φn are the solution of the ti
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