Time-independent Schrödinger equation (TISE)

The eigenvalue equation `-(ħ²/2m) d²ψ/dx² + V(x) ψ = E ψ` describing stationary states with definite energy E in a given potential V(x).

An idealized 1D potential with V=0 in a finite region and V=∞ outside, leading to strictly confined particles and discrete energy levels `E_n ∝ n²`.

A quantum state with energy less than the asymptotic potential, whose wavefunction is normalizable and localized (often with exponential decay at infinity).

The phenomenon where a particle has a nonzero probability to cross a classically forbidden region (E < V), due to the wavefunction penetrating and decaying inside the barrier.

A state with energy below the asymptotic value of the potential, whose wavefunction is normalizable and localized in space.

What is Quantum tunneling?

A purely quantum effect where a particle penetrates and sometimes passes through a potential barrier even when its energy is less than the barrier height.

What is Finite square well?

A 1D potential that is lower inside a region than outside but with finite walls, allowing bound states with exponential tails and a finite number of levels.

What is Probability density?

For a wavefunction ψ(x), the quantity |ψ(x)|², interpreted as the probability per unit length of finding the particle near position x.

Quantum in One Dimension: Boxes, Barriers, and Bound States — Quantum Mechanics Mastery: From Foundations to Modern Frontiers

Q: Which statement is MOST accurate about a particle in a 1D finite square well compared to an infinite well of the same width?

The finite well has a finite number of bound states with energies higher than in the infinite well, and the wavefunctions leak outside the well.. A finite well supports only a finite number of bound states, with energies higher (less negative) than the corresponding infinite-well levels, and the wavefunctions have exponential tails outside. The spectrum is still discrete for bound states. It does not require a specific particle mass.

Q: Transmission coefficient T

The ratio of transmitted probability current to incident current in a scattering problem; in 1D, satisfies R + T = 1 with reflection coefficient R.

Q: What is Bound state?

A state with energy below the asymptotic value of the potential, whose wavefunction is normalizable and localized in space.

Q: What is Quantum tunneling?

A purely quantum effect where a particle penetrates and sometimes passes through a potential barrier even when its energy is less than the barrier height.

Q: What is Finite square well?

A 1D potential that is lower inside a region than outside but with finite walls, allowing bound states with exponential tails and a finite number of levels.

Q: What is Probability density?

For a wavefunction ψ(x), the quantity |ψ(x)|², interpreted as the probability per unit length of finding the particle near position x.

Setting the Stage: 1D Quantum Potentials

Why 1D Quantum?

We study particles in one spatial dimension to understand how boxes, wells, and barriers shape quantum behavior, from confinement to tunneling.

The TISE

Stationary states satisfy the time-independent Schrödinger equation: `-(ħ²/2m) d²ψ/dx² + V(x) ψ = E ψ`. We solve this in regions of constant `V(x)`.

Boundary Conditions

For finite potentials, `ψ(x)` and its derivative `dψ/dx` must be continuous at boundaries. These conditions quantize energy in bound systems.

Why It Matters Now

1D models still guide modern nanotechnology: quantum wells, nanowires, and tunneling devices all rely on these basic ideas.

Infinite Square Well: Building the Box

Defining the Box

Infinite square well: `V(x)=0` for `0<x<L`, and `V(x)=∞` elsewhere. The particle is strictly confined between `x=0` and `x=L`.

Inside the Well

For `0<x<L`, TISE becomes `d²ψ/dx² + k² ψ = 0` with `k² = 2mE/ħ²`. General solution: `ψ(x) = A sin(kx) + B cos(kx)`.

Boundary at x=0

Because `V=∞` outside, `ψ(0)=0`. This forces `B=0`, leaving `ψ(x) = A sin(kx)` inside the well.

Boundary at x=L

`ψ(L)=0` gives `A sin(kL)=0`. For nonzero `A`, `sin(kL)=0`, so `kL = nπ`, with `n=1,2,3,...`.

Infinite Well Energies and Normalized States

Energy Quantization

From `kn = nπ/L`, energies are `En = (ħ²/2m)(nπ/L)²`. Energies are discrete, with `E_n ∝ n²`, and no `n=0` state.

Normalizing ψ

We require `∫₀ᴸ |ψn(x)|² dx = 1`. With `ψn = An sin(nπx/L)`, the integral gives `|An|² L/2 = 1`.

Normalized Eigenfunctions

Solving for `An` yields `An = √(2/L)`. So `ψ_n(x) = √(2/L) sin(nπx/L)` for `n=1,2,3,...`.

Physical Meaning

`|ψ_n(x)|²` is the probability density. Higher `n` has more nodes. This underlies real quantum wells in modern semiconductor devices.

Thought Exercise: Probability in the Box

Use this step to practice interpreting `|ψ(x)|^2`.

Consider the ground state `n = 1` in an infinite well from `0` to `L`:

`ψ_1(x) = √(2/L) sin(πx/L)`.

a. Without integrating, decide: is the particle more likely to be near the center (`x ≈ L/2`) or near the walls (`x ≈ 0` or `L`)? Why?

b. Sketch (mentally or on paper) `ψ1(x)` and `|ψ1(x)|^2`. Mark the points where each is zero and where they are maximum.

Now consider the first excited state `n = 2`:

`ψ_2(x) = √(2/L) sin(2πx/L)`.

a. How many nodes (zeros inside `0 < x < L`) does `ψ_2(x)` have?

b. In which regions (roughly) is the particle most likely to be found? Compare to the ground state.

Challenge yourself: For general `n`, how many nodes does `ψ_n(x)` have inside the well? How might this relate to the idea of higher "energy" states having more oscillations?

Pause for 2–3 minutes to reason through these questions before moving on.

Worked Example: Energy Levels in a Nanowire

Physical Setup

Electron in a 1D infinite well of length `L = 5.0 nm`. Use `E_n = (ħ²/2m)(nπ/L)²` to find `E₁` and `E₂`.

Compute Constants

`ħ²/2m ≈ 6.11×10⁻³⁹ J·m²`, and `(π/L)² ≈ 3.95×10¹⁷ m⁻²` for `L = 5.0×10⁻⁹ m`.

Ground State Energy

`E₁ ≈ (6.11×10⁻³⁹)(3.95×10¹⁷) ≈ 2.41×10⁻²¹ J ≈ 0.015 eV` after converting from joules.

First Excited State

Since `E_n ∝ n²`, `E₂ = 4E₁ ≈ 0.060 eV`. These meV-scale spacings are relevant in modern nanodevices.

Finite Wells and Bound States

Finite Square Well

Now the walls are finite: `V(x)=0` for `|x|<a` and `V(x)=V₀` for `|x|≥a`. The particle is mostly inside but can leak out.

Inside vs Outside

Inside: sinusoidal `ψ` with `d²ψ/dx² + k²ψ=0`. Outside (for `E<V₀`): exponential `ψ` with `d²ψ/dx² - κ²ψ=0`.

Bound States

Bound states have `E<V₀` and decaying tails `ψ ∝ e^{-κ|x|}`. Only a finite number of such states exist for a given `V₀` and `a`.

Conceptual Takeaway

Finite walls mean the wavefunction can leak out. This leakage underlies quantum tunneling and makes the model more realistic.

Barriers and Tunneling: Qualitative Picture

Square Barrier Setup

Barrier: `V(x)=V₀` for `0<x<a`, and `V(x)=0` elsewhere. A particle comes from the left with energy `E`.

Classical vs Quantum

Classically, `E<V₀` means total reflection. Quantum mechanically, there is still a nonzero transmission probability `T`.

Wavefunction Regions

Left: incoming+reflected waves. Inside barrier: exponentials. Right: transmitted wave. Coefficients set by continuity.

Reflection and Transmission

Define `R` and `T` from probability currents; they satisfy `R+T=1`. For `E<V₀`, `T` shrinks with higher or wider barriers.

Quick Simulation: Tunneling Through a Square Barrier

Use Python and SciPy to compute the transmission coefficient `T(E)` for a square barrier. This is a simple numerical experiment you can run on your own machine or in an online notebook.

The code below:

Sets up a barrier of height `V0` and width `a`.
Uses an analytic formula for `T(E)` in the `E < V0` regime.
Plots `T` as a function of `E`.

```python

import numpy as np

import matplotlib.pyplot as plt

Physical constants (SI units)

hbar = 1.055e-34 # J·s

m_e = 9.11e-31 # kg (electron mass)

Barrier parameters

V0_eV = 0.5 # barrier height in eV

V0 = V0_eV * 1.602e-19 # convert to J

a = 1e-9 # barrier width = 1 nm

Energy range (E < V0)

E_eV = np.linspace(0.01, 0.49, 300) # in eV

E = E_eV * 1.602e-19 # in J

Compute kappa and transmission T for E < V0

kappa = np.sqrt(2 m_e (V0 - E)) / hbar

Approximate transmission for a single square barrier (E < V0)

T ≈ 1 / (1 + (V02 * np.sinh(kappa * a)2) / (4 E (V0 - E)))

See standard QM textbooks for derivation.

sinh_term = np.sinh(kappa * a)

T = 1.0 / (1.0 + (V02 * sinh_term2) / (4 E (V0 - E)))

plt.figure(figsize=(6,4))

plt.plot(E_eV, T)

plt.xlabel('Energy E (eV)')

plt.ylabel('Transmission T')

plt.title('Tunneling through a 0.5 eV, 1 nm barrier')

plt.grid(True)

plt.tight_layout()

plt.show()

```

Try modifying:

`a` (barrier width) to see how thicker barriers suppress tunneling.
`V0_eV` (barrier height) to see how higher barriers affect `T(E)`.
`m_e` to a smaller effective mass to mimic a semiconductor electron and compare tunneling probabilities.

Check Understanding: Boxes, Wells, and Barriers

Answer this question to check your grasp of 1D bound states and tunneling.

Which statement is MOST accurate about a particle in a 1D finite square well compared to an infinite well of the same width?

The finite well has the same discrete energy levels as the infinite well but shifted by a constant.
The finite well has a finite number of bound states with energies higher than in the infinite well, and the wavefunctions leak outside the well.
The finite well has a continuous energy spectrum and no bound states because the walls are not infinite.
The finite well has bound states only if the particle mass is exactly equal to the electron mass.

Show Answer

Answer: B) The finite well has a finite number of bound states with energies higher than in the infinite well, and the wavefunctions leak outside the well.

A finite well supports only a finite number of bound states, with energies higher (less negative) than the corresponding infinite-well levels, and the wavefunctions have exponential tails outside. The spectrum is still discrete for bound states. It does not require a specific particle mass.

Key Terms Review

Use these flashcards to review the main ideas from this module.

Time-independent Schrödinger equation (TISE): The eigenvalue equation `-(ħ²/2m) d²ψ/dx² + V(x) ψ = E ψ` describing stationary states with definite energy E in a given potential V(x).
Infinite square well: An idealized 1D potential with V=0 in a finite region and V=∞ outside, leading to strictly confined particles and discrete energy levels `E_n ∝ n²`.
Bound state: A quantum state with energy less than the asymptotic potential, whose wavefunction is normalizable and localized (often with exponential decay at infinity).
Quantum tunneling: The phenomenon where a particle has a nonzero probability to cross a classically forbidden region (E < V), due to the wavefunction penetrating and decaying inside the barrier.
Transmission coefficient T: The ratio of transmitted probability current to incident current in a scattering problem; in 1D, satisfies R + T = 1 with reflection coefficient R.
Nodes of a wavefunction: Points where the wavefunction ψ(x) is exactly zero. In the infinite well, the n-th energy eigenstate has n−1 nodes inside the well.

Key Terms

Bound state: A state with energy below the asymptotic value of the potential, whose wavefunction is normalizable and localized in space.
Quantum tunneling: A purely quantum effect where a particle penetrates and sometimes passes through a potential barrier even when its energy is less than the barrier height.
Finite square well: A 1D potential that is lower inside a region than outside but with finite walls, allowing bound states with exponential tails and a finite number of levels.
Probability density: For a wavefunction ψ(x), the quantity |ψ(x)|², interpreted as the probability per unit length of finding the particle near position x.
Infinite square well: A 1D potential that is zero within a finite interval and infinite outside, producing perfectly confined particles and discrete energy eigenvalues.
Reflection coefficient: A dimensionless quantity R giving the fraction of incident probability current that is reflected by a potential barrier or step.
Transmission coefficient: A dimensionless quantity T giving the fraction of incident probability current that emerges on the far side of a barrier.
Time-independent Schrödinger equation: The differential equation `-(ħ²/2m) d²ψ/dx² + V(x) ψ = E ψ` that determines stationary states with definite energy in a given potential.