By letting t=t, show that the solution to the Schrödinger equation is given by xt=xe^(-iEt/) and satisfies the Time-independent Schrödinger Equation.
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Step 1: Start with the time-dependent Schrödinger equation: iħ∂ψ/∂t = -ħ^2/2m ∂^2ψ/∂x^2 + V(x)ψ Show more…
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(a) Write down the one-dimensional time-dependent Schrödinger equation for the wavefunction Ψ(x,t) when the potential energy is V(x). Using a wavefunction of the form Ψ(x,t) = ψ(x) e^(-iEt/ħ), show that this 1D equation can be transformed into the time-independent Schrödinger equation (TISE). Explain clearly why solutions of this form are called stationary states. (b) Use the Heisenberg momentum-position uncertainty relation to estimate the momentum and the kinetic energy of a particle trapped in a one-dimensional box of width L. The potential inside the box is zero, and infinite outside. (c) Find the eigenvalues and the associated normalised eigenvectors of the spin-1/2 matrix Sy = (ħ/2)σy, where σy = (0 -i; +i 0). Comment on the eigenvalues obtained. (d) Consider a system of 4 non-interacting fermions with spin 1/2 (two with spin up and two with spin down) in an infinite square well. The energy levels available for each of the particles are En = n^2 E1, where n = 1, 2, 3... and E1 is the energy of the lowest available level. i. What is the total energy of the system of 4 particles in its ground state? Give also the total energy and configuration of the system when in its 2nd excited state. ii. What would the ground state energy of the system be if all 4 particles were bosons rather than fermions? Give reasons for all your answers above. (e) Find the commutator [p̂x, x̂²], using the fact that p̂x = -iħ(∂/∂x) and x̂ is the usual position operator.
David M.
The wavefunction ψ(x) = 2/∙a sin(5π/a x) cos(3π/a x) describes a particle of mass m in an infinite square well potential with x in the range [0,a]. i. Show that the boundary conditions are satisfied. ii. Show that ψ(x) can be expressed as a sum over two eigenstates, ψp(x) and ψq(x), of the infinite potential well as ψ(x) = Aψp(x) + Bψq(x), where A and B are real constants and p and q are integers indicating the quantum number associated with each eigenstate (Tip: remember the normalized wave-functions identified as solutions for the IPW and use trigonometric identities). iii. Calculate the energy expectation value ⟨E⟩, given by ⟨E⟩ = ∫ ψ*(x)Hψ(x)dx where H is the Hamiltonian, and express it as a multiple of the ground state energy for the infinite square well potential. (Tip: remember that the expectation value for an eigenstate is the eigenvalue! Then, you can also use the known integral ∫ sin ax sin bx dx = sin[(a-b)x] / 2(a-b) - sin[(a+b)x] / 2(a+b), (a^2 ≠ b^2) Make sure you show all relevant steps with details on how you get to the solution for each question.
Sri K.
A free particle in quantum mechanics is described by a plane wave $$ \psi_{k}(x, t)=e^{i\left(k x-\left(h k^{2} / 2 m\right) t !\right.} $$ Combining waves of adjacent momentum with an amplitude weighting factor $\varphi(k)$, we form a wave packet $$ \Psi(x, t)=\int_{-\infty}^{\infty} \varphi(k) e^{i\left[k x-\left(\hbar k^{2} / 2 m\right) t\right]} d k . $$ (a) Solve for $\varphi(k)$ given that $$ \Psi(x, 0)=e^{-x^{2} / 2 a^{2}} $$ (b) Using the known value of $\varphi(k)$, integrate to get the explicit form of $\Psi(x, t)$. Note that this wave packet diffuses or spreads out with time.
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