Consider two operators \hat{a} and \hat{b}. the operation \hat{a}\hat{b} |ψ> means first operate on state |ψ> with operator \hat{b} to produce a new state |φ> ≡ \hat{b} |ψ> , then operate on the new state |φ> with operator \hat{a} to produce yet another state |φ'> ≡ \hat{a} |φ> , i. e. |φ'> = \hat{a} |φ> = \hat{a}\hat{b} |ψ> . note that, in general, the order of operation matters, i. e., \hat{a}\hat{b}\neq\hat{b}\hat{a} (aˆ and bˆ do not commute).
(a) using the definition that the hermitian adjoint of aˆ |ψ> is < ψ| \hat{a}^\dagger , prove that (aˆbˆ)^† = bˆ†aˆ† .
(b) suppose that aˆ and bˆ are both hermitian and that they do not commute aˆbˆ \neq bˆaˆ. is the product aˆbˆ hermitian? is the sum of products aˆbˆ + bˆaˆ hermitian?
(c) suppose that aˆ and bˆ are both hermitian and that they commute aˆbˆ = bˆaˆ. is the product aˆbˆ hermitian?
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In a thunderstorm, charge builds up on the water droplets or ice crystals in a cloud. thus, the charge can be considered to be distributed uniformly throughout the cloud. for the purposes of this problem, take the cloud to be a sphere of diameter 1.00 kilometer. the point of this problem is to estimate the maximum amount of charge that this cloud can contain, assuming that the charge builds up until the electric field at the surface of the cloud reaches the value at which the surrounding air breaks down. this breakdown means that the air becomes highly ionized, enabling it to conduct the charge from the cloud to the ground or another nearby cloud. the ionized air will then emit light due to the recombination of the electrons and atoms to form excited molecules that radiate light. in addition, the large current will heat up the air, resulting in its rapid expansion. these two phenomena account for the appearance of lightning and the sound of thunder. take the breakdown electric field of air to be eb=3.00ă—106n/c. part a estimate the total charge q on the cloud when the breakdown of the surrounding air is reached. express your answer numerically, to three significant figures, using ďµ0=8.85ă—10â’12c2/(nâ‹…m2) .

Acylinder with a movable piston contains 11.7 moles of a monatomic ideal gas at a pressure of 1.32×10^5 pa. the gas is initially at a temperature of 300 k. an electric heater adds 43200 j of energy into the gas while the piston moves in such a way that the pressure remains constant. cp=20.79 j k^−1 mol^−1 for a monatomic ideal gas, and that the number of gas molecules is equal to avogadro's number (6.022×10^23) times the number of moles of the gas. (a) what is the temperature of the gas after the energy is added? (b) what is the change in volume of the gas? (c) how much work is done by the gas during this process?

In a heat engine if 1000 j of heat enters the system the piston does 500 j of work, what is the final internal energy of the system if the initial energy was 2000 j? 1. write the equation 2.list out your known variables 3.plug the numbers into the equations 4.solve 5.write your solution statement that includes initial energy and final