I think that it is easy to show that the last sum is not zero because if it was zero that would mean that \phi = \psi_{1} according to the equations
1=\sum_{k}|a_{k}|^{2}
\phi = \sum_{k}a_{k}\psi_{k}
But as the problem statement says, \phi\neq \psi_{1}, so the sum can't be zero. So far, I...
But if the last sum is not zero that mean that there is a mistake somewhere since the purpose is to obtain \int \phi^{*} \hat{H} \phi d\tau>E_{1} right?
Homework Statement
This is the problem 8.10 from Levine's Quantum Chemistry 5th edition:
Prove that, for a system with nondegenerate ground state, \int \phi^{*} \hat{H} \phi d\tau>E_{1}, if \phi is any normalized, well-behaved function that is not equal to the true ground-state wave function...
[SOLVED] Dirac delta function and Heaviside step function
In Levine's Quantum Chemistry textbook the Heaviside step function is defined as:
H(x-a)=1,x>a
H(x-a)=0,x<a
H(x-a)=\frac{1}{2},x=a
Dirac delta function is:
\delta (x-a)=dH(x-a) / dx
Now, the integral:
\int...
Sorry, I made a mistake in the first equation. Instead of
\frac{4|N|^{2}s^{2}}{l(s^{2}-b_{2})}\left(1-(-1)^{n}cos(bl)\right)dp
Must be:
\frac{4|N|^{2}s^{2}}{l(s^{2}-b^{2})^{2}}\left(1-(-1)^{n}cos(bl)\right)dp
[SOLVED] Probability for a momentum value for particle in a box
Homework Statement
This is the problem 7.37 from Levine's Quantum Chemistry Textbook.
Show that, for a particle in a one-dimensional box of length l, the probability of observing a value of p_{x} between p and p+dp is...