Infinite Square Well: Quantized States, Probability, and Classical Limit

All flashcards

Infinite square well

A one-dimensional model with zero potential inside the box and infinite potential outside.

Allowed wave numbers

The wall conditions permit $k_n=n\pi/L$ for positive integers $n$.

Ground-state energy

The lowest allowed energy is $E_1=\pi^2\hbar^2/(2mL^2)$, which is greater than zero.

Expectation value

The average result expected from many measurements on identically prepared systems.

Bohr's correspondence principle

Quantum predictions approach classical results in the appropriate large-quantum-number limit.

Quiz questions

1. For a particle in a one-dimensional infinite square well of width $L$, which wave numbers satisfy both wall conditions?Answer: $k_n=n\pi/L$, where $n$ is a positive integer.
2. Why is the $n=0$ solution excluded from the infinite-well energy states?Answer: It gives a wave function that is zero everywhere and cannot be normalized.
3. What is the energy of the first excited state relative to the ground-state energy $E_1$?Answer: $E_2=4E_1$.
4. For the $n=2$ state in a one-dimensional infinite square well, what is the probability density at the center, $x=L/2$?Answer: It is zero because the wave function has a node there.
5. Why does a single energy eigenstate have a probability density that does not change with time?Answer: Its time-dependent factor is a phase with magnitude one.