A particle of mass m in the infinite square well (of width a...

A particle of mass $m$ in the infinite square well (of width $a$ ) starts out in the left half of the well, and is (at $t=0$ ) equally likely to be found at any point in that region. (a) What is its initial wave function, $\Psi(x, 0)$ ? (Assume it is real. Don't forget to normalize it.) (b) What is the probability that a measurement of the energy would yield the value $\pi^{2} \hbar^{2} / 2 m a^{2}$ ?

A particle of mass $m$ in the infinite square well (of width $a$ ) starts out in the left half of the well, and is (at $t=0$ ) equally likely to be found at any point in that region.
(a) What is its initial wave function, $\Psi(x, 0)$ ? (Assume it is real. Don't forget to normalize it.)
(b) What is the probability that a measurement of the energy would yield the value $\pi^{2} \hbar^{2} / 2 m a^{2}$ ?

Key Concepts

Infinite Square Well

The infinite square well is a textbook quantum mechanics potential where a particle is confined to a region with infinitely high potential barriers at the boundaries. In this system, the particle cannot exist outside the well, and its allowed energy levels and wave functions (stationary states) are determined by solving the Schrödinger equation with boundary conditions that the wave function vanishes at the well’s edges. This problem provides a clear illustration of quantum confinement and quantized energy levels.

Normalization of Wavefunctions

Normalization ensures that the total probability of finding a quantum particle within the entire space is one. For any wave function, the integral of the squared modulus of the wave function over all space must equal one. When a wave function is defined only in a specific region (such as half of the well in this problem), the normalization constant must be chosen so that this condition holds within that region, making the probability distribution correctly represent a physical state.

Expansion in Energy Eigenstates

Any initial wave function in a quantum system can be expressed as a superposition of the complete set of energy eigenstates of that system. These energy eigenstates form an orthonormal basis, and the coefficients in this expansion are found by projecting the initial wave function onto each eigenstate. This expansion is particularly useful because each eigenstate evolves in time with a simple phase factor, and the squared magnitudes of the coefficients provide the probabilities of obtaining the corresponding energy eigenvalues upon measurement.

Quantum Measurement Postulate

In quantum mechanics, the measurement postulate asserts that if a system is prepared in a state that is a superposition of eigenstates, then the probability of measuring a particular eigenvalue is given by the square of the magnitude of the coefficient associated with the corresponding eigenstate in the wave function's expansion. This principle underlies the prediction of measurement outcomes and is used to determine probabilities for observables, such as energy in this problem.

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