A particle is confined by the potential well V(x)= 0 for |x|<a ∞ otherwise Explain (

step 1 Write down the wave functions for the particle in this well. So we're going to use equation 7 .3

A particle is confined by the potential well V(x)= 0 ...

A particle is confined by the potential well $$ V(x)= \begin{cases}0 & \text { for }|x|<a \\ \infty & \text { otherwise }\end{cases} $$ Explain (a) why we can assume that there is a complete set of stationary states with well-defined parity and (b) why to find the stationary states we solve the TISE subject to the boundary condition $\psi(\pm a)=0$. Determine the particle's energy spectrum and give the wavefunctions of the first two stationary states.

A particle is confined by the potential well
$$
V(x)= \begin{cases}0 & \text { for }|x|<a \\ \infty & \text { otherwise }\end{cases}
$$
Explain (a) why we can assume that there is a complete set of stationary states with well-defined parity and (b) why to find the stationary states we solve the TISE subject to the boundary condition $\psi(\pm a)=0$.
Determine the particle's energy spectrum and give the wavefunctions of the first two stationary states.

Key Concepts

Parity and Symmetry

Parity refers to the behavior of a function under spatial inversion (x to -x). In potentials that are symmetric about the origin, such as the infinite square well, the Hamiltonian is invariant under parity transformation. This symmetry allows the eigenfunctions to be classified as either even (symmetric) or odd (antisymmetric), simplifying the problem and reducing the number of independent solutions that need to be considered.

Boundary Conditions in Quantum Mechanics

In quantum systems with potentials that become infinite outside a certain region, the wavefunction must vanish where the potential is infinite. The application of boundary conditions, such as requiring the wavefunction to be zero at the boundaries of the potential well, is crucial for obtaining physically acceptable solutions. These conditions lead to quantization of the energy spectrum because only certain wavefunctions satisfy both the TISE and the imposed boundary conditions.

Infinite Potential Well (Particle in a Box)

The infinite potential well is a fundamental model in quantum mechanics that illustrates the quantization of energy in a confined space. The potential is zero inside the well and infinite outside, which means the particle is strictly confined within the well. This model not only demonstrates the concept of quantized energy levels but also serves as an idealized case for exploring quantum confinement and the role of boundary conditions in determining the properties of the system.

Stationary States and Energy Eigenstates

Stationary states are solutions of the time-independent Schrödinger equation (TISE) that have a well-defined energy eigenvalue. These eigenstates form a complete set, meaning any state of the system can be expressed as a linear combination of these eigenfunctions. This concept is central to quantum mechanics as it provides the framework for understanding the quantized nature of energy in bound systems.

Time-Independent Schrödinger Equation (TISE)

The TISE is a differential equation that provides the modes (eigenfunctions) of the system along with their corresponding energy eigenvalues. Solving this equation with appropriate boundary conditions yields the stationary states. In confined systems such as potential wells, the TISE becomes an eigenvalue problem where only certain discrete energy levels are allowed.

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