You got six out of eight by my count.
sweetwater said:
a all eigenvalues of hermitian operators are real numbers
True. You can easily prove this by considering a Hermitian operator acting on an eigenvector ##Av = \lambda v, A = A^*##. Left-multiplying by ##v^*## gives:
$$v^*Av = \lambda v^*v$$
and taking the conjugate transpose gives:
$$\begin{align}
(v^*Av)^* &= (v^*\lambda v)^* \\
v^*A^*v &= \lambda^*v^*v \\
v^*Av &= \lambda^* v^* v
\end{align}$$
where the fact that ##A## is hermitian was used on the last line. Combining these two results gives ##\lambda^* = \lambda##, which means that ##\lambda## must be real.
sweetwater said:
b a hermitian operator cannot contain the imaginary number i
False. One example is momentum: ##\hat{p} = -i\hbar \nabla##.
sweetwater said:
c an increas in the particle mass would decrease the ground state energy of both the particle in a box and harmonic oscillator
True, since the particle in a box energy is
$$E = \frac{n^2h^2}{8mL^2}$$
and the harmonic oscillator energy is
$$E = \hbar\sqrt{\frac{k}{m}}\left(n+\frac{1}{2}\right)$$
sweetwater said:
d wave function must be real
False. The wavefunction can be complex. In fact, take any stationary state and evolve it in time, and you get a wavefunction with a complex phase:
$$\psi(t) = e^{-iEt/\hbar}\psi(0)$$
sweetwater said:
e if wave function is zero at the nuclesu for all H atom stationary states
False. For s states (with zero angular momentum), the wavefunction is nonzero.
sweetwater said:
f the most probable value of the electorn-nucleus distance in a gorund state H atom is zero
True. ##|\psi|^2 \propto exp(-r)## for ##r \geq 0##. This function reaches its max value for ##r = 0##.
sweetwater said:
g FOr the atom ground sate the electron is confined to move within a sphere of fixed radius
False. ##|\psi|^2## is nonzero for any finite value of ##r##, so the electron has a nonzero probability of being any radial distance from the nucleus.
sweetwater said:
hany photon with energy E photon >or = .75hcRh can casue a hydrogen atom to go form the n=1 state to the n=2 tate
False. The excitation spectrum is discrete, not continuous. Thus, for an excitation to take place, the photon needs to have the exact value of the difference between energy levels.