Rectangular potential barrier

Key takeaways

Rectangular potential barrier belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Rectangular potential barrier to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Rectangular potential barrier from memory before moving on to harder problems.

Reference excerpt

In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left. Although classically a particle behaving as a point mass would be reflected if its energy is less than V 0 {\displaystyle V_{0}} , a particle actually behaving as a matter wave has a non-zero probability of penetrating the barrier and continuing its travel as a wave on the other side. In classical wave-physics, this effect is known as evanescent wave coupling. The likelihood that the particle will pass through the barrier is given by the transmission coefficient, whereas the likelihood that it is reflected is given by the reflection coefficient. Schrödinger's wave-equation allows these coefficients to be calculated.

Calculation

The time-independent Schrödinger equation for the wave function ψ ( x ) {\displaystyle \psi (x)} reads

H ^ ψ ( x ) = [ − ℏ 2 2 m d 2 d x 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) {\displaystyle {\hat {H}}\psi (x)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)\right]\psi (x)=E\psi (x)}

where H ^ {\displaystyle {\hat {H}}} is the Hamiltonian, ℏ {\displaystyle \hbar } is the (reduced) Planck constant, m {\displaystyle m} is the mass, E {\displaystyle E} the energy of the particle and

V ( x ) = V 0 [ Θ ( x ) − Θ ( x − a ) ] {\displaystyle V(x)=V_{0}[\Theta (x)-\Theta (x-a)]}

is the barrier potential with height V 0 > 0 {\displaystyle V_{0}>0} and width a {\displaystyle a} . Θ ( x ) = 0 , x < 0 ; Θ ( x ) = 1 , x > 0 {\displaystyle \Theta (x)=0,\;x<0;\;\Theta (x)=1,\;x>0}

is the Heaviside step function, i.e.,

V ( x ) = { 0 if x < 0 V 0 if 0 < x < a 0 if a < x {\displaystyle V(x)={\begin{cases}0&{\text{if }}x<0\\V_{0}&{\text{if }}0<x<a\\0&{\text{if }}a<x\end{cases}}}

… excerpt ends here. Continue reading the full article.

Illustrations

Rectangular potential barrier illustration

Worked examples

Example 1 — a first encounter with Rectangular potential barrier

Start with the simplest possible case. Write down what Rectangular potential barrier claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rectangular potential barrier before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rectangular potential barrier ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rectangular potential barrier

In research: Rectangular potential barrier appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rectangular potential barrier in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rectangular potential barrier is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum mechanical potentials, Quantum models, Scattering theory, so understanding it makes those chapters shorter.
In everyday life: Look for Rectangular potential barrier outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Rectangular potential barrier in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rectangular potential barrier means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is Rectangular potential barrier in simple terms?

In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. The problem consists of solving the one-dimensional tim…

Why does Rectangular potential barrier matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rectangular potential barrier?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rectangular potential barrier.