(2) By using the recursion formulas for the Hermite polynomials, obtain the expectation value of (x*) in a harmonic oscillator.
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- Let y, (x) denote the orthonormal stationary states of a system corresponding to the energy En. Suppose that the normalized wave function of the system at time t = 0 is µ(x,0) and suppose that a measurement of the energy yields the value E1 with probability 1/2, E2 with probability 3/8, and E3 with probability 1/8. (a) Write the most general expansion for Þ(x,0) consistent with this information. (b) What is the expansion for the wave function of the system at time t, Þ(x, t)?4) [swHW] It turns out that any function that has a finite number of finite-magnitude discontinuities and a finite number of extrema (maximums and minimums) over a finite interval can be represented exactly with an infinite series of cosine and sine functions called a Fourier Series. The conditions that the function must meet, called the "Dirichlet conditions", are not very restrictive, so most functions you will encounter in physics will have an associated Fourier Series. To give you a sense of how this is possible consider a very simple f(x) = x function on the interval -πShow that the average value of x2 in the one-dimensional infinite potential energy well is L2 ((1/3)-(1/2(n^2)(pi^2))).Be *(1) the position operator for a particle subjected to a potential of a one-dimensional harmonic oscillator P mox (Ĥ =+ 2m 2 Evaluate [î(t),î(0)] Heisenberg's chart in(c) Consider a system of two qubits with canonical basis states {|0) , |1)}. Write down an example for a two- qubit density matrix corresponding to a separable pure state and an example for a two-qubit density matrix corresponding to an entangled pure state.(AA) ²( ▲ B) ²≥ ½ (i[ÂÂ])² If [ÂÂ]=iñ, and  and represent Hermitian operators corresponding to observable properties, what is the minimum value that AA AB can have? Report your answer as a decimal number with three significant figures.