Question

1) Consider a 1D particle in a box. The ground state wavefunction for the system is $\psi(x) = N \sin(\frac{\pi x}{L})$ and the energy is $\frac{h^2}{8mL^2}$ from the Hamiltonian $\hat{H} = -\frac{h^2}{2m}\frac{\partial^2}{\partial x^2}$ for $0 \le x \le L$. $N$ is the normalization constant. Answer the following questions based off this system. a) Consider the trial wavefunction $\phi(x) = N[c \cos(\frac{\pi x}{L}) + d \sin(\frac{\pi x}{L})]$ where $a$, $b$ are variational parameters. Write out the expression to determine the value of the energy, $<E>$. Substitute in for the wavefunction and Hamiltonian. DO NOT SOLVE ANY INTERGRALS. (4 points)

          1) Consider a 1D particle in a box. The ground state wavefunction for the system is
$\psi(x) = N \sin(\frac{\pi x}{L})$ and the energy is $\frac{h^2}{8mL^2}$ from the Hamiltonian $\hat{H} = -\frac{h^2}{2m}\frac{\partial^2}{\partial x^2}$ for
$0 \le x \le L$. $N$ is the normalization constant. Answer the following questions based
off this system.
a) Consider the trial wavefunction $\phi(x) = N[c \cos(\frac{\pi x}{L}) + d \sin(\frac{\pi x}{L})]$ where $a$, $b$ are
variational parameters. Write out the expression to determine the value of the
energy, $<E>$. Substitute in for the wavefunction and Hamiltonian. DO NOT SOLVE
ANY INTERGRALS. (4 points)

1) Consider a 1D particle in a box. The ground state wavefunction for the system is
ψ(x) = N sin((π x)/(L)) and the energy is (h^2)/(8mL^2) from the Hamiltonian Ĥ = -(h^2)/(2m)(∂^2)/(∂ x^2) for
0 ≤ x ≤ L. N is the normalization constant. Answer the following questions based
off this system.
a) Consider the trial wavefunction ϕ(x) = N[c cos((π x)/(L)) + d sin((π x)/(L))] where a, b are
variational parameters. Write out the expression to determine the value of the
energy, <E>. Substitute in for the wavefunction and Hamiltonian. DO NOT SOLVE
ANY INTERGRALS. (4 points)

Added by Enrique W.

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