In quantum mechanics, if a wave function is an eigenfunction of an operator (such as momentum, energy, etc.) then the wave function will yield a definite eigenvalue of that operator when measured. For example, if p^ψ (x) = pψ(x), then the wave function ψ(x) is a momentum-eigenfunction that represents a particle whose momentum will be measured to be a definite eigenvalue p. (Note that the Schr¨odinger equation is simply the equation that yields energy eigenfunction/eigenvalues: H^ψ(x) = Eψ(x), where E is the energy eigenvalue of a particle whose wave function is ψ(x)). Wave functions that are not eigenfunctions of an operator represent particles that will not yield definite eigenvalues of that operator, and can only be determined probabilistically.(a) Show whether or not the wave function ψ(x) = Ae^(ipx/h-bar) of a free particle is a momentum-eigenfunction with a definite momentum.(b) Is this same wave function an energy eigenfunction? If so, what is its energy? Explain.(c) Show whether or not the wave function un(x) = √(2/L) sin (nπx/L) of a free particle is a momentum eigenfunction with a definite momentum.(d) Is this same wave function an energy eigenfunction? If so, what is its energy?(e) A superposition state is a wave function that is a linear combination of energy eigenfunctions, e. g. the wave function given by a linear sum of wave functions of the form in(d): ψ(x) = 1/(√2) u1(x) +1/ (√2) u2(x). Show whether or not ψ(x) itself is an energy eigenfunction with a definite energy.

What is special about a dc circuit ?

Question 1 what is amperage? is the rate of doing work. is the rate of flow of protons in electric current. represents the amount of pressure behind electron flow. is the rate of flow of electrons in electric current. 2 points question 2 what is voltage? is the rate of doing power. represents the amount of pressure behind electron flow. is the rate of doing work. is the rate of flow of electrons in electric current. 2 points question 3 what is power? is the rate of flow of protons in electric current. is the rate of flow of electrons in electric current. is the rate of doing work. represents the amount of pressure behind electron flow. 2 points question 4 if we multiply volts times amps we get what? power circuit work current 2 points question 5 what are two ways alternating currents are similiar? in both ac and dc electrons flow in the same pattern. in both ac and dc, the flow of electrons changes directions back and forth. both ac and dc are only possible in certain materials with atoms that will allow electron flow. both ac and dc involve the flow of electrons. 4 points question 6 how does the flow of electrons flow in an alternating current? the flow of electrons is always slower in an alternating current than within a direct current. the flow of electrons is not constant and forward; it changes direction back and forth. electrons flow from from a higher affinity to that of a lower affinity. electron flow is constant and only in a forward direction. 2 points question 7 what is the flow like in a direct current? the flow of electrons is not constant and forward; it changes direction back and forth. the flow of electrons is constant and only in a forward direction. the flow of electrons go from a higher affinity to a lower affinity. the flow of electrons are always faster in a direct current. 2 points question 8 how is an electric current able to flow? electrons flow from the higher affinity to lower affinity and electrical current is generated. protons flow from the higher affinity to lower affinity and electrical current is generated. the movement of protons from one atom to another leads to an electric charge. the movement of electrons from one atom to another atom in a line results in a flow of electric current. 2 points question 9 how do electrons move from the two different types of metal in a battery? protons flow from the metal with the lower affinity to the metal with higher affinity and electrical current is generated. electrons flow from the metal with the lower affinity to the metal with higher affinity and electrical current is generated. electrons flow from the metal with the higher affinity to the metal with lower affinity and electrical current is generated. protons flow from the metal with the higher affinity to the metal with lower affinity and electrical current is generated.

Ahot-air balloon takes off from the ground traveling vertically with a constant upward acceleration of magnitude g/4. after time interval δt, a crew member releases a ballast sandbag from the basket attached to the balloon. how many seconds does it take the sandbag to reach the ground? (express your answer in terms of δt)