The normalised wavefunction for an atom with atomic mass A=96, magnetically trapped in a 1D harmonic oscillator of frequency 690 Hz can be written: ψ=(0.179 ψ0)+(0.107 i ψ5)+(h ψ9). As the individual wavefunctions are orthonormal, use your knowledge to work out |h|, and hence find the expectation value for the energy of the atom, in peV. This is at the opposite end of the energy spectrum to the LHC!

Consider a quantum particle with energy E approaching a potential barrier of width L and heightV0 > E from the right (as shown in image). The wavefunction of the particles in the region x > L is given by ψ = A exp {−i (kx + ωt)} , where A, k and ω are all constants. Use the Gamow factor formalism to calculate an approximate expression for the transmission rate of these particles through the barrier.

(Requires integral calculus.) Imagine that a quanton's wavefunction at a given time is y(x) Ae-x/al, where A is an unspecified = constant and a = 35 nm . If we were to perform an experiment to locate the quanton at this time, what would be the probability (as a percent) of a result within ±0.47 a = ±16.45 nm of the origin? The probability is Note: Round the final answer to one decimal place. %.

The normalised wavefunction for an electron in an infinite 1D potential well of length 80 pm can be written:ψ=(0.587 ψ2)+(0.277 i ψ7)+(g ψ6). As the individual wavefunctions are orthonormal, use your knowledge to work out |g|, and hence find the expectation value for the energy of the particle, in eV.

Explain each step

The wave function of a particle at time t= 0 is given by w(0) = (4,) +|u2})), where |u,) and u,) the normalized eigenstates with eigenvalues E and E, are respectively, (E, > E, ). The shortest time after which y(t) will become orthogonal to |w(0)) is - ħn (а) 2(E, – E,) (b) E, - E, (c) E, - E, (d) E, - E,