HW 8: Superposition States and Time Dependence - Quantum Mechanics for Chemistry

Question-1¶

Write down a wavefunction for particle in a box which is in a superposition state of first three levels with equal coefficients.
Write down average energy of such a state
Write down time dependent wave function for such a state

Write the parabolic $f(x)=x^2$ as a linear superposition of orthogonal eigenfunctions of particle in a box over the interval $0\leq x \leq L$ .

f(x)=\sum^{n=\infty}_{n=1} b_n sin \frac{n\pi x}{L}

Using the orthogonality show that coefficients can be determined via $b_n=\frac{2}{L} \int^{L}_0 f(x) sin \frac{n\pi x}{L}$ . You may want to make use of one of the integrals from the table provided in the previous HW.
Write down the expansion of $f(x)$ in terms of the explicitely defined coefficients $b_n$ .

Decompose the following vectors and functions in terms of the respective orthogonal components.

$|v> = (3,4,5,6)$ Decompose in terms of orthogonal unit vectors:
- unit vectors: $|1>=(1,0,0,0)$ , $|2>=(0,1,0,0)$ etc.
$f(x) = 4x+2x^3+5$ Decompose in terms of the orthognal functions:
- $|1>=1$ , $|2>=x$ and $|3>=3x^3-1$ defined over interval $-1\leq x\leq +1$

Take the dot products.
- $<2| = (0,0,0,1)$ and $<w|=(1,2,3,4)$
- $<2|=x$ and $<w|=1+x+x^3$

State of the quantum cat in the box is described as $|alive>$ and $|dead>$ orhtogonal states. You do 100 experiments with 100 identically prepared boxes and find that in 80 boxes the time cat is/was alive. Write down the ket |cat> that will describe the state of the quantum cat in the box.
Measuring the energy of a particle in 1D box we find that 40% of the time the particle is found in the ground state, the 40% time in the first excited staates and the 20% of time in the fifth excited state. Write down the normalized wavefunction that will describe such a state.

A particle of mass m in an infinite potential well of length a has the following initial wave function at t =0:

|\psi(0)>=\frac{2}{\sqrt{7a}} sin \frac{\pi x}{a}+\sqrt{\frac{6}{7a}} sin\frac{2\pi x}{a}+\frac{2}{\sqrt{7a}} sin \frac{3\pi x}{a}

Verify the normalization $<\psi | \psi>=1$
If we start measuring the energy, what values will we be obtaining in an experiment and with what probabilities?
What would be the average energy at t=0?
Write the wave function at a later time t?
Determine the probability of finding the particle at a time t in a state $|\phi(t)>=\frac{1}{\sqrt{a}}sin(\frac{3\pi x}{a})e^{-iE_3 t/\hbar}$ by computing the projection $<\phi(t) |\psi(t)>$
What would be the average energy at a later time t?