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- By employing the prescribed definitions of the raising and lowering operators pertaining to the one-dimensional harmonic oscillator: x = ħ 2mω -(â+ + â_) hmw ê = i Compute the expectation values of the following quantities for the nth stationary staten. Keep in mind that the stationary states form an orthogonal set. 2 · (â+ − â_) [ pm 4ndx YmVndx = 8mn a. The position of particle (x) b. The momentum of the particle (p). c. (x²) d. (p²) e. Confirm that the uncertainty principle is satisfied for all values of n11. Evaluate (r), the expectation value of r for Y,s (assume that the operator f is defined as "multiply by coordinate r).Why does (r) not equal 0.529 for Y,,? In this problem,use 4ardr = dt.A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: (x,0) = A [v₁ (x) + ₂(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy, if you exploit the orthonormality of 1 and 2. Recall that, having normalized at t = 0, you can rest assured that it stays normalized—if you doubt this, check it explicitly after doing part (b).) (c) (b) Find (x, t) and (x, t)|². Express the latter as a sinusoidal function of time, as in Example 2.1. To simplify the result, let w = ²ħ/2ma². Compute (x). Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a /2, go directly to jail.) (d) Compute (p). (As Peter Lorre would say, "Do it ze kveek vay, Johnny!”) (e) If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them? Find the expectation value of H. How does…