Paper 1, Section II, A

Applications of Quantum Mechanics | Part II, 2015

Define the Rayleigh-Ritz quotient R[ψ]R[\psi] for a normalisable state ψ|\psi\rangle of a quantum system with Hamiltonian HH. Given that the spectrum of HH is discrete and that there is a unique ground state of energy E0E_{0}, show that R[ψ]E0R[\psi] \geqslant E_{0} and that equality holds if and only if ψ|\psi\rangle is the ground state.

A simple harmonic oscillator (SHO) is a particle of mass mm moving in one dimension subject to the potential

V(x)=12mω2x2V(x)=\frac{1}{2} m \omega^{2} x^{2}

Estimate the ground state energy E0E_{0} of the SHO by using the ground state wavefunction for a particle in an infinite potential well of width aa, centred on the origin (the potential is U(x)=0U(x)=0 for x<a/2|x|<a / 2 and U(x)=U(x)=\infty for x>a/2)|x|>a / 2). Take aa as the variational parameter.

Perform a similar estimate for the energy E1E_{1} of the first excited state of the SHO by using the first excited state of the infinite potential well as a trial wavefunction.

Is the estimate for E1E_{1} necessarily an upper bound? Justify your answer.

[\left[\right. You may use : π/2π/2y2cos2ydy=π4(π261)\int_{-\pi / 2}^{\pi / 2} y^{2} \cos ^{2} y d y=\frac{\pi}{4}\left(\frac{\pi^{2}}{6}-1\right) \quad and ππy2sin2ydy=π(π2312)]\left.\quad \int_{-\pi}^{\pi} y^{2} \sin ^{2} y d y=\pi\left(\frac{\pi^{2}}{3}-\frac{1}{2}\right) \cdot\right]

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