Find a bound state by matching a finite boundary and its tail
Take a symmetric one-dimensional well with zero potential inside absolute x less than a0 and height V0 outside. A bound state has energy strictly between zero and V0 in this well-bottom reference. For the lowest even state, the interior amplitude is A cos(kx), with k=sqrt(2mE)/hbar. Outside it decays as A cos(ka0)exp[−kappa(|x|−a0)], where kappa=sqrt[2m(V0−E)]/hbar.
Learning outcome: Derive the matching equation from both continuity conditions. Retain the initial branch bracket and numerical residual for two depths, compare energy against the infinite-wall value and independently integrate the exponential tails. State what remains uncomputed about higher states.
Starting recap
Review definite integrals, complex amplitudes, probability moments, elementary differential equations and energy-momentum units. State boundary and normalization conditions, and distinguish calibration from unused validation.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
At z0=2, bisection gives z≈1.029866529. Thus E/V0≈0.265156268 and kappa a0≈1.714460537. Normalizing the interior and both tails gives exterior probability about 0.097682859. A nonzero exterior density is compatible with a stationary bound state and zero current.
Open the experimentPredict
What does a nonzero tail of this real stationary bound state imply?
Experiment
Derive the matching equation from both continuity conditions. Retain the initial branch bracket and numerical residual for two depths, compare energy against the infinite-wall value and independently integrate the exponential tails. State what remains uncomputed about higher states.
Bracket a finite-well bound state and normalize its tails
Worked teaching model · synthetic values · Symmetric finite well V0 outside |x|<a0 and zero inside. z0=a0 sqrt(2mV0)/hbar. Solve only the lowest even state z tan z=sqrt(z0²−z²),0<z<min(z0,pi/2), with80 bounded bisection steps. Energy reference is well bottom; tail probability is not an outward flux in this real stationary state.
Reset keeps your prediction, captures and practice history.
Normalized bound-state density including tails
Graph axes rescale independently; read units and values, not screen angles. 1: lowest even state (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Normalized bound-state density including tails
lowest even state
| u = x/a0 | density per unit u |
|---|---|
| -3 | 0.0001760447 |
| -2.9 | 0.0002480499 |
| -2.8 | 0.0003495064 |
| -2.7 | 0.0004924604 |
| -2.6 | 0.0006938849 |
| -2.5 | 0.0009776955 |
| -2.4 | 0.001377589 |
| -2.3 | 0.001941046 |
| -2.2 | 0.002734967 |
| -2.1 | 0.003853614 |
| -2 | 0.005429807 |
| -1.9 | 0.007650688 |
| -1.8 | 0.01077995 |
| -1.7 | 0.01518913 |
| -1.6 | 0.02140173 |
| -1.5 | 0.0301554 |
| -1.4 | 0.04248946 |
| -1.3 | 0.05986837 |
| -1.2 | 0.08435553 |
| -1.1 | 0.1188584 |
| -1 | 0.1674734 |
| -0.9 | 0.2276288 |
| -0.8 | 0.2915118 |
| -0.7 | 0.3564216 |
| -0.6 | 0.4196141 |
| -0.5 | 0.478418 |
| -0.4 | 0.5303471 |
| -0.3 | 0.5732064 |
| -0.2 | 0.6051837 |
| -0.1 | 0.6249274 |
| 0 | 0.6316027 |
| 0.1 | 0.6249274 |
| 0.2 | 0.6051837 |
| 0.3 | 0.5732064 |
| 0.4 | 0.5303471 |
| 0.5 | 0.478418 |
| 0.6 | 0.4196141 |
| 0.7 | 0.3564216 |
| 0.8 | 0.2915118 |
| 0.9 | 0.2276288 |
| 1 | 0.1674734 |
| 1.1 | 0.1188584 |
| 1.2 | 0.08435553 |
| 1.3 | 0.05986837 |
| 1.4 | 0.04248946 |
| 1.5 | 0.0301554 |
| 1.6 | 0.02140173 |
| 1.7 | 0.01518913 |
| 1.8 | 0.01077995 |
| 1.9 | 0.007650688 |
| 2 | 0.005429807 |
| 2.1 | 0.003853614 |
| 2.2 | 0.002734967 |
| 2.3 | 0.001941046 |
| 2.4 | 0.001377589 |
| 2.5 | 0.0009776955 |
| 2.6 | 0.0006938849 |
| 2.7 | 0.0004924604 |
| 2.8 | 0.0003495064 |
| 2.9 | 0.0002480499 |
| 3 | 0.0001760447 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Lowest even dimensionless wave number | 1.029867 | ratio |
| Final numerical lower bracket | 1.029867 | ratio |
| Final numerical upper bracket | 1.029867 | ratio |
| Matching residual | -1.110223e-15 | ratio |
| Energy divided by barrier height | 0.2651563 | ratio |
| Energy in hbar squared over 2m half-width squared units | 1.060625 | ratio |
| Infinite-wall ground comparison in same units | 2.467401 | ratio |
| Probability outside the well | 0.09768286 | ratio |
| Probability inside the well | 0.9023171 | ratio |
| Decay length divided by half-width | 0.5832739 | ratio |
| Density at selected outside coordinate | 0.005429807 | per unit x/a0 |
Finite confinement has a normalized evanescent tail and an energy below its infinite-wall counterpart.
Continuity of psi and its derivative at x=a0 gives k tan(ka0)=kappa. With z=ka0 and z0²=z²+(kappa a0)² this becomes z tan z=sqrt(z0²−z²). The lowest even branch is monotone between0 and min(z0,pi/2). Its normalization denominator is1+sin(2z)/(2z)+cos²z/(kappa a0). The exterior probability is the last term divided by that denominator. The real stationary tail has zero probability current, not a steady escaping population.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Finite confinement has a normalized evanescent tail and an energy below its infinite-wall counterpart.
Continuity of psi and its derivative at x=a0 gives k tan(ka0)=kappa. With z=ka0 and z0²=z²+(kappa a0)² this becomes z tan z=sqrt(z0²−z²). The lowest even branch is monotone between0 and min(z0,pi/2). Its normalization denominator is1+sin(2z)/(2z)+cos²z/(kappa a0). The exterior probability is the last term divided by that denominator. The real stationary tail has zero probability current, not a steady escaping population.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which state, calibration or boundary premise is independently justified, and what unused observation could distinguish an alternative?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.