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Science · Advanced Physics
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3. (a) Explain what is meant by an observable in quantum mechanics. Consider an observable represented by an operator A, with eigenvalue equation Auaun in the usual notation. Explain the physical meaning of the eigenvalues a and eigenfunctions u, The normalised eigenfunctions of operators representing observables form an orthonormal set state mathematically what this means 4 marks) (b) The wavefunction of a hydrogen atom at a certain time is given by -Ar expl- sin cos where A is a constant and ao is the Bohr radius Given this wavefunction, what are the possible results of a measurement of (i) The energy of the electron? (ii) The orbital angular momentum of the electron? (iii) The z-component of the orbital angular momentum of the electron? (iv) In all three cases, what is the probability of getting each possible measurement result? 4 marks) (c) Write down an expression to calculate the mean distance of the electron from the nuclear centre for the ground state of hydrogen. (2 marks)

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