Quatum mechanics basic theory questions

[sweetwater]

Homework Statement

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

Homework Equations

none

The Attempt at a Solution

a true
bfasle
ctrue
dfalse
efalse
f true
gtrue
h true

 

[TeethWhitener]

You got six out of eight by my count.

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.

b a hermitian operator cannot contain the imaginary number i

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

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
$$E = \hbar\sqrt{\frac{k}{m}}\left(n+\frac{1}{2}\right)$$

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

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.

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 $r = 0$.

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.

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.