For a one-dimensionał harmonic oscillator, introduction of the dimen-sionless coordinate and encrgy variables y = r(mwo/hi)'/² and en =2E,/ liwo gives a Schrödinger cquation with kinetic encrgy operator Tand potential energy V = y².dy(a) Using the fact that the only non-vanishing «lipole matrix element is(7n + 1]y|1) = /t1 (and its Heruuitcan conjugate), find values for all thenon-vanishing inatrix elements of y that connect to the ground state |0).(b) The oscillator is perturbed by an harmonic potential V'Find the correction to the ground state energy in the lowest non-vanishingorder. (If you did not get complete answers in part(a), leave your result interms of clearly defined natrix clements, etc.)ay³.

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For a one-dimensional harmonic oscillator, introduction of the dimen- sionless coordinate and energy variables y = r(mwo/hi)'/2 and en = 2E,/ liwo gives a Schrödinger cquation with kinetic energy operator T = and potential energy V = y2. dy (a) Using the fact that the ouly non-vanishing «lipole matrix element is (n + 1|y|n) = /tl (and its Heruuitean conjugate), find values for all the non-vanishing inatrix elements of y that connect to the ground state |0). (b) The oscillator is perturbed by an harmonic potential V' Find the correction to the ground state energy in the lowest non-vanishing order. (If you did not get complete answers in part(a), leave your result in terms of clearly defined matrix clements, ctc.) ay.

For a one-dimensional harmonic oscillator, introduction of the dimen- sionless coordinate and energy variables y = r(mwo/hi)'/2 and en = 2E,/ liwo gives a Schrödinger cquation with kinetic energy operator T = and potential energy V = y2. dy (a) Using the fact that the ouly non-vanishing «lipole matrix element is (n + 1|y|n) = /tl (and its Heruuitean conjugate), find values for all the non-vanishing inatrix elements of y that connect to the ground state |0). (b) The oscillator is perturbed by an harmonic potential V' Find the correction to the ground state energy in the lowest non-vanishing order. (If you did not get complete answers in part(a), leave your result in terms of clearly defined matrix clements, ctc.) ay.