Quantum Bound States: A Complete Guide to Particle Confinement

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Here is something that should bother you: trap an electron in a tiny box and try to measure its momentum. You will find it moving, sometimes quite fast, even though you never gave it any energy. The particle cannot sit still. Ever. Not even at absolute zero.

This is not a measurement error or some experimental quirk. It is a fundamental feature of nature that Heisenberg's uncertainty principle demands. Confine a particle tightly and you automatically give it kinetic energy. The universe does not care about our classical intuition here (and frankly, classical physics gets quantum bound states spectacularly wrong).

What makes this genuinely remarkable is that the particle's energy cannot take just any value. Pluck a guitar string and you hear specific notes, not a continuous smear of frequencies. Quantum particles work the same way. Only certain "notes" (energy levels) are allowed. This quantization emerges naturally when you solve Schrodinger's equation with the constraint that the wave function must behave properly at the boundaries. No additional assumptions needed.

How to Use This Simulation

Potential Type Selection

Use the four buttons at the top to switch between different quantum systems:

Main Controls

Quantum Number (n): Click buttons 1-5 to select which energy eigenstate to display. Higher n means more nodes in the wave function and higher energy. You can also press keys 1-5 on your keyboard.

Well Parameters:

• Width L: Adjust the well width (0.1 to 5 nm). Wider wells have lower energy levels.
• Mass: Change the particle mass in units of electron mass. Heavier particles have lower energy levels.
• V₀ (Finite/Double Well only): Set the barrier height in eV.

Display Options:

• Show |ψ|²: Toggle the probability density (purple shaded area)
• Show V(x): Toggle the potential energy curve (orange line)
• Show Nodes: Highlight points where the wave function crosses zero

Superposition Mode: Check this box to create a superposition of multiple states. Then click multiple n-buttons to combine them.

Header Buttons (Top Right)

Bottom Tabs

• Equations: View the governing equations for the current potential type
• Compare: See how quantum and classical predictions differ
• Constants: Reference table of physical constants used
• Quiz: Test your understanding with interactive questions

Tips for Exploration

1. Start with the Infinite Well at n=1, then click through n=2, 3, 4, 5 to see how the wave function develops more nodes
2. Switch to Finite Well and notice how the wave function "leaks" outside the well boundaries
3. Try Harmonic Osc. and observe that energy levels are equally spaced (unlike the square well)
4. Use Double Well to see tunneling - notice how the ground state has probability in BOTH wells

The Physics of Bound States

The Time-Independent Schrodinger Equation

What Schrodinger figured out in 1926 was that particles are described by wave equations, just like sound or light. The time-independent version looks like this:

-hbar^2/(2m) d^2 psi/dx^2 + V(x) psi = E psi

Looks intimidating? Here is what it actually says: the way the wave function curves (left side, first term) plus the potential energy contribution (second term) must equal the total energy times the wave function (right side). The elegant part is that this simple balance gives you everything. All the weird quantum behavior falls out of these symbols.

If you have tried solving differential equations, you know most have infinitely many solutions. Not this one. Only certain values of E produce wave functions that do not blow up at infinity or violate boundary conditions. Those special E values are the quantized energy levels.

Infinite Square Well: The Particle in a Box

Here is the physics textbook's favorite toy problem, and there is a good reason for that. Trap a particle between perfectly rigid walls (at x = 0 and x = L) and ask: what wave functions are allowed?

Think about a guitar string fixed at both ends. When you pluck it, you do not get arbitrary vibrations. You get standing waves. The ends must stay still, which forces the wavelength to fit into the string length as lambda = 2L/n. Same physics here, different context.

The wave functions are:

psi_n(x) = sqrt(2/L) sin(n pi x / L)

And here is what happens when you actually try this (well, mathematically): the energy eigenvalues follow a quadratic dependence:

E_n = n^2 pi^2 hbar^2 / (2 m L^2)

Notice that n-squared factor. Double the quantum number and you quadruple the energy. The gaps between levels get wider as you climb the energy ladder. But the truly strange part? The ground state (n=1) has nonzero energy. The particle cannot stop moving. Ever.

Key Features of the Infinite Well

Finite Square Well: Reality Check

Infinite walls are mathematically convenient but physically absurd. No real barrier is infinitely high. What happens when you use a realistic potential with depth V_0?

Here is what happens when you actually try this: the wave function refuses to stop at the wall. It leaks into the classically forbidden region where E < V_0, decaying exponentially but definitely nonzero. Classically, a particle without enough energy to climb over the barrier should never be found there. Quantum mechanically, it absolutely is.

This leakage has consequences. The effective wavelength inside the well stretches longer than in the infinite case, which means lower energies for all the bound states. And unlike the infinite well (which has infinitely many bound states no matter how shallow you make it), the finite well supports only a limited number:

Number of bound states approximately equals sqrt(2 m V_0 L^2 / hbar^2) / pi

The exact energies come from solving transcendental equations, the kind that make you reach for a computer:

tan(kL/2) = kappa/k (even states) -cot(kL/2) = kappa/k (odd states)

where k = sqrt(2mE)/hbar and kappa = sqrt(2m(V_0 - E))/hbar. No closed-form solutions exist. You iterate numerically or plot both sides and look for intersections.

Quantum Harmonic Oscillator

If you could only study one quantum system, pick the harmonic oscillator. Any potential looks like a parabola near its minimum (Taylor expand and see for yourself). Molecular vibrations, photons in a cavity, phonons in solids, quantum field theory itself: they all reduce to harmonic oscillators when you squint.

The energy levels are:

E_n = (n + 1/2) hbar omega

Look at that formula. Two features should jump out. First, the levels are equally spaced by hbar omega. Unlike the square well where gaps widen as you go up, here every step costs the same energy. Second (and this is the profound part), even the ground state with n=0 has energy hbar omega / 2. That is the zero-point energy again, and it has been measured experimentally through the Casimir effect: two parallel metal plates in vacuum feel an attractive force because they modify the quantum vacuum fluctuations between them.

The wave functions involve Hermite polynomials (those polynomials you probably saw once and forgot):

psi_n(x) = (alpha / pi)^(1/4) / sqrt(2^n n!) H_n(alpha x) exp(-alpha^2 x^2 / 2)

where alpha = sqrt(m omega / hbar). The ground state is a Gaussian: maximum probability at the center, trailing off symmetrically. If you could slow this down enough to watch, you would see probability density concentrated where a classical oscillator would spend the most time (the turning points), but only at high quantum numbers. Low quantum numbers look nothing like classical behavior.

Double Well and Tunneling

Put a barrier between two potential wells. Classical physics says a particle stuck in the left well stays in the left well (unless you give it enough energy to climb over). Quantum mechanics laughs at this.

The wave function does not stop at the barrier. It decays exponentially through the forbidden region, but if the barrier is thin enough, some probability amplitude leaks through to the other side. The particle can be found in either well without ever going over the barrier. What Feynman called "the only mystery of quantum mechanics" happens right here.

If you had two separate wells, they would have identical ground state energies. Connect them through tunneling and the degeneracy breaks. You get symmetric and antisymmetric combinations with slightly different energies:

Delta E approximately equals E_0 exp(-2 integral of kappa dx)

where the integral is over the barrier region. The thinner or lower the barrier, the larger the splitting.

This tunnel splitting is not just a textbook curiosity:

• Ammonia inversion: the basis for the first atomic clock (1949)
• Proton transfer in DNA: how hydrogen bonds in base pairs shuffle around
• Molecular isomerization: chemical reactions that proceed through barriers rather than over them

Exploration Activities

Activity 1: Energy Level Scaling

Objective: Verify the n^2 energy dependence in the infinite square well.

Here is a prediction you can test right now: if energy scales as n-squared, then E_2 should be exactly 4 times E_1. Not approximately. Exactly.

Steps:

1. Select "Infinite Well" potential
2. Set L = 1 nm, m = 1 m_e
3. Click through n = 1 to 5, recording each energy
4. Calculate the ratios E_2/E_1, E_3/E_1, E_4/E_1, E_5/E_1
5. Verify these ratios are approximately 4, 9, 16, 25

Do you see perfect squares? That is quantum mechanics telling you something profound about confinement.

Activity 2: Finite vs Infinite Well Comparison

Objective: Observe how finite barriers lower energy levels.

Real barriers are never infinite. What does that cost you in terms of energy?

Steps:

1. Set up infinite well: L = 1 nm, n = 1. Record E_1.
2. Switch to finite well with V_0 = 50 eV. Record new E_1.
3. Reduce V_0 to 20 eV, then 10 eV. Record energies.
4. At what V_0 does the first excited state cease to be bound?
5. Enable "Show probability" to see penetration into barriers.

Watch that exponential tail. The wave function does not care that classical physics says "forbidden."

Activity 3: Harmonic Oscillator Zero-Point Energy

Objective: Explore the ground state energy of the quantum harmonic oscillator.

The zero-point energy scales linearly with frequency. Stiffer springs (higher omega) mean more ground state energy. Can you verify this?

Steps:

1. Select "Harmonic Osc." potential
2. Note the ground state energy at omega = 1 x 10^15 rad/s
3. Double omega to 2 x 10^15 rad/s. By what factor does E_0 change?
4. Compare energy level spacing (E_2 - E_1) vs (E_3 - E_2)
5. Observe the Gaussian shape of the wave function

Equal spacing. This is the only potential that does this. Every other shape gives unequal gaps.

Activity 4: Tunnel Splitting in Double Well

Objective: Observe energy level splitting from quantum tunneling.

You cannot see tunneling directly, but you can measure its consequences: nearly degenerate levels that split apart.

Steps:

1. Select "Double Well" potential
2. Set barrier height to 10 eV, L = 1 nm
3. Compare E_1 and E_2 energies (should be nearly degenerate)
4. Reduce barrier height to 5 eV. Does splitting increase or decrease?
5. Use superposition mode to see symmetric/antisymmetric combinations

Lower barriers mean more tunneling which means larger splitting. The math and the simulator agree.

Measurement Collapse

Here is where quantum mechanics gets genuinely weird. Before you look, the particle exists as a spread-out wave function, with probability distributed across space. After you look? Definite position. Somewhere specific.

The simulation lets you experience this. Click "Measurement Collapse," then click anywhere on the wave function. The probability of finding the particle at position x is proportional to |psi(x)|^2 (higher where the wave function is bigger). After measurement, the wave function collapses to a localized state. The smooth wave becomes a spike.

This collapse is instantaneous. Non-local. Non-unitary. It breaks every rule that governs normal quantum evolution. Einstein hated it. Schrodinger was disturbed by it. The measurement problem remains one of the deepest puzzles in physics, ninety years and counting.

Classical vs Quantum Comparison

Click the classical comparison toggle. What you see should surprise you.

For a classical particle bouncing in a box:

• It spends equal time everywhere at the same speed (mostly uniform probability)
• Near the walls it slows down and turns around (slightly higher probability there)
• Energy can be anything you want (continuous spectrum)
• Position and momentum are both defined simultaneously, always

For a quantum particle:

• Probability varies according to |psi|^2, with wiggly peaks and valleys
• Probability is exactly zero at infinite walls (the wave function is zero there, period)
• Only discrete energies are allowed (try getting E = 1.5 E_1, you cannot)
• Position and momentum cannot both be precisely known (uncertainty principle)

At high quantum numbers, the quantum probability starts resembling the classical one. The correspondence principle at work. But at low n? Classical physics gives you the wrong answer entirely.

Physical Constants Reference

Challenge Questions

Test your understanding. Some of these are straightforward if you have been paying attention. Some will make you think.

Level 1: Conceptual

1. Why must the wave function be zero at the walls of an infinite square well? (Hint: what happens to the kinetic energy term if psi is nonzero where V is infinite?)

2. What physical quantity determines whether levels are equally spaced (harmonic oscillator) or quadratically spaced (square well)? (Think about the shape of the potential.)

3. Why does a finite well have fewer bound states than an infinite well of the same width? (Where does that extra wave function penetration go?)

Level 2: Analytical

1. An electron is confined to a 1 nm infinite square well. Calculate:

   • Ground state energy (use E_1 = pi^2 hbar^2 / 2mL^2)
   • Energy required to excite from n=1 to n=2
   • Wavelength of photon needed for this transition

2. For a quantum harmonic oscillator with hbar omega = 0.1 eV, what is:

   • Zero-point energy
   • Energy of the n=10 state
   • Energy difference between adjacent levels (should be the same for all transitions, yes?)

Level 3: Advanced

1. Derive the condition for a finite square well to have exactly one bound state. (Start with the transcendental equation and think about when the second solution appears.)

2. For a symmetric double well with barrier height V_b and width a, estimate the tunnel splitting of the ground state doublet using the WKB approximation. (The integral through the barrier matters.)

3. A particle in a box has wave function psi(x,0) = A[psi_1(x) + psi_2(x)]. Find:

   • Normalization constant A (orthonormality helps here)
   • Time evolution psi(x,t)
   • Period of probability density oscillation (watch for the beating frequency)

Common Misconceptions

These errors show up constantly, even among students who should know better. Worth reviewing.

Frequently Asked Questions

Why are only certain energies allowed?

Same reason a guitar string only vibrates at certain frequencies. The wave function must satisfy boundary conditions (zero at walls, decaying to zero at infinity) and remain normalizable. Most values of E give solutions that blow up or misbehave. Only special E values work, and those become your quantized levels [1].

What is the physical meaning of the wave function?

Born's interpretation (1926): |psi(x)|^2 gives the probability density for finding the particle at position x when measured. The wave function itself is not directly observable. You never see psi. You only see the statistics of measurement outcomes [2].

How do bound states relate to atomic orbitals?

Atomic orbitals are the same physics in three dimensions with the Coulomb potential (1/r instead of a square well). The discrete energy levels of hydrogen (which produce those sharp spectral lines that puzzled physicists before Bohr) are bound states in the nuclear potential [3].

Can a particle tunnel out of a bound state?

Yes, if the walls are not infinite. This is radioactive alpha decay: the alpha particle is bound inside the nucleus but can tunnel through the potential barrier to escape. For deeply bound states the tunneling probability is exponentially small, which is why some nuclei have billion-year half-lives [4].

What determines how many bound states exist?

For a finite well: depth V_0 and width L. Deeper wells trap more states. Wider wells do too. The formula N approximately equals sqrt(2mV_0L^2/hbar^2)/pi gives a rough count. An infinite well, of course, has infinitely many bound states at all depths [5].

References

1. Griffiths, D.J. "Introduction to Quantum Mechanics," 3rd Edition, Cambridge University Press, 2018. Chapters 2-3.

2. MIT OpenCourseWare 8.04: Quantum Physics I. Available at: https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/

3. Shankar, R. "Principles of Quantum Mechanics," 2nd Edition, Springer, 1994. Chapters 5-7.

4. Cohen-Tannoudji, C., Diu, B., and Laloe, F. "Quantum Mechanics," Wiley, 1977. Volume 1.

5. NIST: Fundamental Physical Constants. Available at: https://physics.nist.gov/cuu/Constants/

6. HyperPhysics: Particle in a Box. Georgia State University. Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html

7. The Feynman Lectures on Physics: Quantum Mechanics, Volume III. Available at: https://www.feynmanlectures.caltech.edu/III_toc.html

8. OpenStax University Physics: Quantum Mechanics. Available at: https://openstax.org/books/university-physics-volume-3/pages/7-4-the-quantum-particle-in-a-box

9. Zettili, N. "Quantum Mechanics: Concepts and Applications," 2nd Edition, Wiley, 2009.

10. Gasiorowicz, S. "Quantum Physics," 3rd Edition, Wiley, 2003.

About the Data

Physical constants come from NIST CODATA 2022 (the current standard):

• Reduced Planck constant: hbar = 1.054571817 x 10^-34 J s
• Electron mass: m_e = 9.1093837015 x 10^-31 kg
• Elementary charge: e = 1.602176634 x 10^-19 C

Where possible, we use exact analytical solutions. The infinite square well has closed-form energies and wave functions. So does the harmonic oscillator. The finite well requires numerical root-finding for the transcendental equations. The double well uses a variational approximation that captures the tunnel splitting correctly.

How to Cite

Simulations4All Team. "Quantum Bound States Simulator: Interactive Exploration of Particle Confinement." Simulations4All, 2025. https://simulations4all.com/simulations/quantum-bound-states-simulator

For academic use:

@misc{simulations4all_quantum_bound_states, title = {Quantum Bound States Simulator}, author = {Simulations4All Team}, year = {2025}, url = {https://simulations4all.com/simulations/quantum-bound-states-simulator} }

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