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what is the potential energy inside the potential well for a particle in potential well problem?

Inside the (infinite) potential well, the potential energy is taken to be zero ; all the energy eigenvalues you compute are therefore purely kinetic energy of the particle.

Quick Scoop

When people talk about the “particle in a potential well” (the classic quantum mechanics problem), they usually mean one of two standard models:

  1. Infinite square well (particle in a box)
    • The potential energy function is

V(x)={0,0<x<L∞,x≤0 or x≥LV(x)= \begin{cases} 0,&0<x<L\\ \infty,&x\le 0\text{ or }x\ge L \end{cases}V(x)={0,∞,​0<x<Lx≤0 or x≥L​

so inside the well (the region where the particle can actually exist), the potential energy is exactly zero.

 * The allowed energies EnE_nEn​ you learn and memorize (like En∝n2/L2E_n\propto n^2/L^2En​∝n2/L2) are therefore the particle’s **kinetic** energies, since E=K+VE=K+VE=K+V and V=0V=0V=0 inside.
  1. Finite square well
    • Here the potential might be written as

V(x)={−V0,∣x∣<a0,∣x∣≥aV(x)= \begin{cases} -V_0,&|x|<a\\ 0,&|x|\ge a \end{cases}V(x)={−V0​,0,​∣x∣<a∣x∣≥a​

or some similar convention. Inside the well the potential energy is a constant (often chosen as −V0-V_0−V0​), lower than the outside region.

 * In that case, the total energy EEE of a bound state lies between the bottom of the well and the outside region, and **inside** the well you still have E=K+VE=K+VE=K+V, with VVV just a fixed number −V0-V_0−V0​, so the kinetic energy is K=E−(−V0)=E+V0K=E-(-V_0)=E+V_0K=E−(−V0​)=E+V0​.

Mini breakdown

  • In the infinite well :
    • Potential energy inside: V =0V=0V=0.
* Total energy levels: purely kinetic.
  • In a finite well :
    • Potential energy inside: a constant (often −V0-V_0−V0​), lower than outside.
* The shape (square, smooth, etc.) can vary, but inside the “flat” region we still treat VVV as constant.
  • Physically:
    • The “well” is just a region where the potential energy is lower than outside.
    • The exact number you assign (0, −V0-V_0−V0​, etc.) is a choice of reference ; only differences in potential energy matter for the physics.

One-sentence template answer for exams

For the standard infinite potential well (particle in a box), the potential energy inside the well is taken to be zero, so the quantized energy levels are purely kinetic; in a finite well, the potential inside is a constant lower value (often written as −V0-V_0−V0​) relative to the outside.

Meta description (SEO-style):
Learn what the potential energy inside the potential well is for the classic “particle in a box” problem in quantum mechanics, how it’s defined in infinite and finite wells, and why we usually set it to zero by convention.

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