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1D motion of a quantum particle

Consider the 1D motion of a quantum particle in an external potential with its total energy given by the equation in the attachment.

(a) Explain why according to quantum mechanics, the ground state energy (i.e. the lowest possible energy of the particle) can never be zero.
(b) By making use of a certain fundamental particle of quantum mechanics,
(i) show that the ground state energy of the particle has an order of magnitude given by the equation in the attachment.
(ii) show explicitly that the quantity given in the RHS of the above expression does indeed have a physical dimension equal to that of energy.

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This solution provides step-by-step equations and explanations for questions involving the concept of ground state energy.

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