The Time-Independent Schrödinger Equation
Energy conservation written for a wavefunction gives a differential equation whose solutions are the allowed stationary states. Solving a quantum problem means solving that equation with boundary conditions.
What a learner can do afterwards
- Identifies kinetic, potential and total energy terms in the equation
- States the conditions a physical solution has to satisfy at a boundary
- Explains what makes a state stationary when the particle is still described by a wave
1 · Read
Classical energy conservation says total energy is momentum squared over 2 m plus potential U. The time-independent Schrodinger equation writes the same budget for a wave: a curvature term for kinetic energy, plus U times psi for potential, equals E times psi. Solving a quantum problem means finding the psi and the E values that satisfy this equation with the boundary conditions.
Inside an infinite well U is zero and the walls are impenetrable, so the equation becomes a pure curvature equation. The walls demand psi equals zero at both ends, which only sine waves with whole half-waves between the walls can meet. Each fitting wave carries its own allowed energy.
A stationary state has one fixed energy, and its probability density never changes in time. The particle is not sitting still: it stays spread in a wave whose shape of chances is frozen. This separated form applies when the potential itself does not change with time.
Read the three terms before calculating: curvature means kinetic, U psi means potential, E psi means total. Check the walls first, since they pick the solutions. Reject n equals zero: it is zero everywhere and cannot be normalised.
Each term of the equation names an energy, the walls select the solutions, and stationary states freeze the chances in time.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.