In the region 0 ≤x ≤a, a particle is described by the wave function ψ1(x)=-b(x^2-a^2). In the re

step 1 So we have a particle in the box question and this is a continuity question. We have to find a c

In the region 0 ≤x ≤a, a particle is described by the wave...

In the region $0 \leq x \leq a$, a particle is described by the wave function $\psi_{1}(x)=-b\left(x^{2}-a^{2}\right)$. In the region $a \leq x \leq w,$ its wave function is $\psi_{2}(x)=(x-d)^{2}-c .$ For $x \geq w, \psi_{3}(x)=0 .(a)$ By applying the continuity conditions at $x=a,$ find $c$ and $d$ in terms of $a$ and $b .(b)$ Find $w$ in terms of $a$ and $b$.

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Key Concepts

Boundary Conditions in Piecewise Problems

Many quantum systems have potentials defined in segments, leading to solutions that are valid only in specified regions. The application of boundary conditions—both for the function and its derivative—is essential in connecting these segmental solutions and in determining relationships between constants in the overall solution.

Continuity of the Derivative

If the potential is well-behaved (not singular), the first derivative of the wave function is also required to be continuous at the boundaries. This ensures that the kinetic energy operator, which involves second derivatives of the wave function, does not introduce inconsistencies, and it leads to additional equations that help determine the unknown parameters in the solution.

Continuity of the Wave Function

In quantum mechanics, the wave function must be continuous across the boundaries between regions, ensuring that the probability density does not exhibit any unphysical jumps. This condition is applied at the interfaces of piecewise-defined potentials to connect the different solutions smoothly.

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