Consider a quantum state V = 1/2-1 + √/2₂₁ + 1/24/2 that is a superposition of three eigenstates of operator with eigenvalues w_₁ = -1, W1 = 1, and w2 = 2 (same as subscripts k in above). The expectation value of is w/1 √ 1 √3+2√2-1 √6
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- Suppose you have an observable N with three eigenvalues 4, 8, and -1, with orthonormal eigenvectors |1), 2), and 3), respectively. A quantum particle in state |) = |1) – iv3 *|2) + 3). Use this information to V5 3 answer the following questions. What is the probability that you measure 8? 3 V3 3The eigenstates of the particle-in-a-box are written, n = √ sin (™T). If L = 10.0, what is the expectation value for the quantity 2ħ² + p² in the n = 3 eigenstate? Report your answer as a multiple of ħ². (Note: ô = −iħª) d dx'A proton is confined in box whose width is d = 750 nm. It is in the n = 3 energy state. What is the probability that the proton will be found within a distance of d/n from one of the walls? Include a sketch of U(x) and ?(x). Sketch the situation, defining all your variables
- (a) Consider the following wave function of Quantum harmonic oscillator: 3 4 V(x, t) = =Vo(x)e¯iEot +. Where, Eo, E, are the energy values corresponding to the ground state and the first excited state. Show that the expectation value of î in this state is periodic in time. What is the period? (b) Consider a quantum harmonic oscillator. The operator â4 is defined by : mw 1 â4 = 2h d: 2ħmw Find the expectation value of Hamiltonian for the state â4,(x). ma 1/4,- x2 and , (x) = - mw ma, x2 [Given ,(x) = () mw1/4 2mw x: 1. πή2i+1 i+1 |- +> + 3 [recall, |+ -> means that particle #1 is in the |+> state (usual Z basis) and #2 is in the |-> state.] A) Show that this state is already normalized. B) Is this state separable or entangled? C) A measurement of S, is made on particle #1. What are the possible results and with what probabilities? D) A measurement of Sz is made on particle #2. What are the possible results and with what probabilities? E) Calculate the expectation value of the correlation function between these two measurements . (Don't use matrices -- use probabilities!)Consider an electron in a one-dimensional, infinitely-deep, square potential well of width d. The electron is in the ground state. (a) Sketch the wavefunction for the electron. Clearly indicate the position of the walls of the potential well on your sketch. (b) Briefly explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.