Homework Help: Quatum mechanics basic theory questions

1. Feb 21, 2009

sweetwater

1. The problem statement, all variables and given/known data
this is true and false questions
a all eigenvalues of hermitian operators are real numbers
b a hermitian operator cannot contain the imaginary number i
c an increas in the particle mass would decrease the ground state energy of both the particle in a box and harmonic oscillator
d wave function must be real
e if wave function is zero at the nuclesu for all H atom stationary states
f the most probable value of the electorn-nucleus distance in a gorund state H atom is zero
g FOr the atom ground sate the electron is confined to move within a sphere of fixed radius
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

2. Relevant equations
none

3. The attempt at a solution
a true
bfasle
ctrue
dfalse
efalse
f true
gtrue
h true

2. Jan 29, 2017

TeethWhitener

You got six out of eight by my count.
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.

False. One example is momentum: $\hat{p} = -i\hbar \nabla$.

True, since the particle in a box energy is
$$E = \frac{n^2h^2}{8mL^2}$$
and the harmonic oscillator energy is
$$E = \hbar\sqrt{\frac{k}{m}}\left(n+\frac{1}{2}\right)$$

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:
$$\psi(t) = e^{-iEt/\hbar}\psi(0)$$

False. For s states (with zero angular momentum), the wavefunction is nonzero.

True. $|\psi|^2 \propto exp(-r)$ for $r \geq 0$. This function reaches its max value for $r = 0$.

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.

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.