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What must be the velocity of a spacecraft if 1 ℎ on its clock is to correspond to 1 ℎ + 1 𝑠 on a clock on
the earth?
(i) Compute the value of 𝛾 for a particle traveling at half the speed of light. Give your answer
to three significant figures.
(ii) Determine the energy required to give an electron a speed of 0.90 that of light, starting from
rest.
If A and B are Hermitian operators, show that AB is Hermitian if and
only if A commutes with B
If A and B are Hermitian operators, show that AB is Hermitian if and only if A commutes with B
Let A and B be two non-commuting Hermitian operators. Determine
which of the following operators are Hermitians:
(a) AB
(b) [A, B]
(c) {A, B} = AB + BA
(d) ABA
(e) An where n is an integer.
If A, B and C are Hermitian operators, determine if the following combinations are Hermitian: (a) A + B (b) 1 2i [A, B] (c) (ABC − CBA) (d) A2 + B2 + C 2 (e) (A + iB)
Using Rayleigh-Jeans formula, find the total energy density. Can you explain Stefan-Boltzmann
law from this? Explain your answer.
A Hermitian operator Aˆ has only three normalized eigenfunctions ψ1, ψ2, ψ3, with corresponding eigenvalues a1 = 1, a2 = 2, a3 = 3, respectively. For a particular state Φ of the system, there is a 50% chance that a measure of A produces a1 and equal chances for either a2 or a3. (a) Calculate hAi. (b) Express the normalized wave function Φ of the system in terms of the eigenfunctions of Aˆ.
If the real normalized functions f(x) and g(x) are not orthogonal, show that their sum f(x) +g(x) and their difference f(x)−g(x) are orthogonal.
Show that if the linear operators Aˆ and Bˆ do not commute, the operators (AˆBˆ + BˆAˆ) and i[A, ˆ Bˆ] are Hermitian.
