At t=0 the particle of Problem 5.1 has the wavefunction ψ(x)= 1 / √(2 a) for |x|<a 0

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At t=0 the particle of Problem 5.1 has the wavefunction...

At $t=0$ the particle of Problem $5.1$ has the wavefunction $$ \psi(x)= \begin{cases}1 / \sqrt{2 a} & \text { for }|x|<a \\ 0 & \text { otherwise }\end{cases} $$ Find the probabilities that a measurement of its energy will yield: (a) $9 \hbar^{2} \pi^{2} /\left(8 m a^{2}\right) ;$ (b) $16 \hbar^{2} \pi^{2} /\left(8 m a^{2}\right)$

At $t=0$ the particle of Problem $5.1$ has the wavefunction
$$
\psi(x)= \begin{cases}1 / \sqrt{2 a} & \text { for }|x|<a \\ 0 & \text { otherwise }\end{cases}
$$
Find the probabilities that a measurement of its energy will yield: (a) $9 \hbar^{2} \pi^{2} /\left(8 m a^{2}\right) ;$ (b) $16 \hbar^{2} \pi^{2} /\left(8 m a^{2}\right)$

Key Concepts

Quantum Measurement and Projection Postulate

According to the principles of quantum mechanics, when a measurement is made, the system 'collapses' to one of the eigenstates of the observable being measured. The probability of the system collapsing to a particular eigenstate is proportional to the square of the amplitude obtained by projecting the initial state onto that eigenstate.

Expansion in Energy Eigenstates

A general state of a quantum system can be expressed as a linear combination (or superposition) of energy eigenstates. This expansion allows one to calculate the probability of measuring specific energy values by determining the contribution (or amplitude) of each eigenstate in the overall state.

Normalization

Normalization is the process of ensuring that the total probability of finding the particle over all space is one. This is achieved by adjusting the wavefunction so that the integral of its absolute square over its domain equals one, which is fundamental for correctly predicting measurement outcomes.

Wavefunction

The wavefunction in quantum mechanics is a complex function that encapsulates all information about a quantum system. It is used to calculate probabilities for the outcomes of any measurement performed on the system by taking its absolute square, provided the wavefunction is properly normalized.

Energy Eigenstates

Energy eigenstates are the specific solutions to the time-independent Schrödinger equation for a system, each corresponding to a definite energy value (eigenvalue). These states form a complete basis that can describe any state of the system when appropriately superposed.

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