3. (a) What is the physical significance of the expectation value? (b) According to the measurement...

 

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3. (a) What is the physical significance of the expectation value? (b) According to the measurement postulate of quantum mechanics, what happens if Q is measured, followed immediately by a measurement of a compatible quantity and then a further measurement of Q? (You may assume that all the eigenvalues of Q and are different.) (c) Consider the harmonic oscillator potential, V(x) = wfmx/2. The first two eigenfunctions are po(x) = = ()* V(x) = -x/2 where a = mwo/h. Write down a general expression for the energy eigenvalues, [2] and explain the physical significance of the term wo in classical physics. (d) Assuming that the system in part (c) is in the ground state, calculate (i) the expectation value of potential energy, (v). (ii) the expectation value of kinetic energy, (). (x) = - Comment on the physical interpretation of your answers for (i) and (ii). (Note, some integrals are supplied at the end of the question over page that you may find useful.) , $1(x) = (e) Now consider that an infinite potential barrier is placed at x = 0, confining the particle to the region x > 0 such that the potential becomes 160\/ 40 (10) + x 0 x>0. wmx/2 By considering the boundary conditions that a physically valid wave function must satisfy, explain why the new ground state, (x) has the form xe e-x/2 xe -x/2 x 0 x > 0. [1] [2] [2] [2] [1] [4] 3. (a) What is the physical significance of the expectation value? (b) According to the measurement postulate of quantum mechanics, what happens if Q is measured, followed immediately by a measurement of a compatible quantity and then a further measurement of Q? (You may assume that all the eigenvalues of Q and are different.) (c) Consider the harmonic oscillator potential, V(x) = wfmx/2. The first two eigenfunctions are po(x) = = ()* V(x) = -x/2 where a = mwo/h. Write down a general expression for the energy eigenvalues, [2] and explain the physical significance of the term wo in classical physics. (d) Assuming that the system in part (c) is in the ground state, calculate (i) the expectation value of potential energy, (v). (ii) the expectation value of kinetic energy, (). (x) = - Comment on the physical interpretation of your answers for (i) and (ii). (Note, some integrals are supplied at the end of the question over page that you may find useful.) , $1(x) = (e) Now consider that an infinite potential barrier is placed at x = 0, confining the particle to the region x > 0 such that the potential becomes 160\/ 40 (10) + x 0 x>0. wmx/2 By considering the boundary conditions that a physically valid wave function must satisfy, explain why the new ground state, (x) has the form xe e-x/2 xe -x/2 x 0 x > 0. [1] [2] [2] [2] [1] [4]

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a The expectation value in quantum mechanics represents the average value of a physical observable in a given quantum state It provides a statistical ... View the full answer