A particle in a 1D box of length a has trial function $\varphi=x(a-x)$ on 0≤x≤a. Find percent error in its ground-state variational energy, to 2 decimals. Exact energy is $h^2/(8ma^2)$ and $\int_0^a\varphi^*\varphi dx=a^5/30$.
The second derivative is φ″ = −2. The kinetic-energy numerator is $-(\hbar^2/2m)\int\varphi\varphi^{\prime\prime}dx=(\hbar^2/m)\int_0^ax(a-x)dx=\hbar^2a^3/(6m)$. Dividing by a⁵/30 gives $E_{var}=5\hbar^2/(ma^2)$. Since $E_0=\pi^2\hbar^2/(2ma^2)$, relative error is $100(10/\pi^2-1)=1.32\%$. It is positive, consistent with the variational bound.
More Structure: Postulates of quantum mechanics. Operators. Commutation relations and expectations values; the role of the wave function and its absolute squares, time dependent and time independent Schrödinger equations. Born interpretation. Dirac bra-ket notation. Examples of model solvable quantum mechanical systems. questions
GATE 2019 CY Q43– A one-dimensional system has =(kx)e3ikx. Which is a possible result of a momentum…
GATE 2024 CY Q63– 1 and 2 are normalized eigenfunctions of a Hermitian operator. =3i 1+2 2 and =-2i 1+5 2.…
GATE 2025 CY Q28– Two noninteracting particles are confined to a one-dimensional infinite box. Their…
GATE 2025 CY Q60– The kinetic energies of an electron (e) and a proton (p) are E and 3E, respectively. Given…
GATE 2026 CY Q27– For =x(a-x), with constant a, which statement(s) is/are correct?
GATE 2026 CY Q61– Two atoms have spherical coordinates (1,/2,/2) and (1,/4,3/2), with distance in Å and…