Show, by the same method used on in this chapter, that the kinetic-energy operator on…

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Observables, commutators and the uncertainty relations

Problem

Show, by the same method used on

\hat p

in this chapter, that the kinetic-energy operator on its own,

\hat T = \hat p^2/2m = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2}

, is Hermitian: that

\int f^*(\hat Tg)\,dx = \int(\hat Tf)^*g\,dx

for normalisable

f

and

g

that, with their first derivatives, go to zero at

x \to \pm\infty