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2) With this notation, Heisenberg’s expression reads [r, p] = i . 3) is often said to be the most fundamental equation of quantum mechanics. The prescription for the momentum operator, Eq. 5), is a solution to this equation if we take the x component of the position operator to be multiplication by x. To see this, just substitute rx = x and px = ( / i)∂/∂x in Eq. 3) and evaluate the results when the commutator operates on an arbitrary function ψ: [x, px ] ψ = xpx ψ − px xψ = x =x ∂ψ − i ∂x i ∂ψ i ∂x − i ∂(xψ) ∂x x∂ψ +ψ =− ψ=i ∂x i ψ.

15) represents a function with real and imaginary parts, both of which oscillate in time at a frequency of Ek / 2π , or Ek / h. The constant term ζ in the exponential is a phase shift that depends on our choice of zero time. As long as we are considering only a single particle, ζ will not affect any measurable properties of the system and we can simply set it to zero. (We will return to this point for systems containing many particles in Chap. ) Combining Eqs. 15) and setting ζ = 0 gives a complete wavefunction for state k: Ψk (r, t) = ψk (r) exp(−iEk t/ ) .

1 Superposition States The eigenfunctions of the Hamiltonian operator can be shown to form a complete set of orthogonal functions. By “orthogonal” we mean that the product of any two different members of such a set (Ψi and Ψk ), when integrated over all space, is zero: Ψi|Ψk = 0 . 19) The functions sin x and sin(2x), for example, are orthogonal. A general property of eigenvalue equations such as the time-independent Schrödinger equation is that if ψi is a solution, then so is the product of ψi with any constant.