Home › Physics › Modern Physics › Motion of a Particle in a One-Dimensional Step Potential
Modern Physics
Motion of a Particle in a One-Dimensional Step Potential
Motion of a quantum mechanical particle in a step potential involves solving the time-independent Schrödinger equation across a discontinuous potential energy boundary. It demonstrates the fundamental wave nature of matter, resulting in quantum reflection and transmission even when the particle energy exceeds the potential step height.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of potential steps, total energy versus potential energy comparisons, and the probability of reflection and transmission.
PGT extension
Rigorous matching of wavefunctions and their first derivatives at the boundary, derivation of reflection and transmission coefficients, and behavior of particles with energy less than or greater than the potential height.
Prerequisites
Time-independent Schrödinger equation in one dimension
Boundary conditions for quantum mechanical wavefunctions
Euler's formula for complex exponential functions
Definition of Step Potential
A step potential is defined mathematically as V(x) = 0 for x < 0 and V(x) = V0 for x ≥ 0.
The potential energy changes discontinuously at the boundary x = 0.
It divides space into two distinct regions: Region I (x < 0) and Region II (x ≥ 0).
Case I: Particle Energy Greater than Potential Step (E > V<sub>0</sub>)
In Region I, the particle wave is a combination of incident and reflected plane waves.
In Region II, the particle wave consists solely of a transmitted wave moving in the positive x-direction.
Unlike classical mechanics where a particle slows down but never reflects, quantum mechanics predicts a non-zero probability of reflection.
Case II: Particle Energy Less than Potential Step (E < V<sub>0</sub>)
In Region I, the total wavefunction is a standing wave formed by incident and totally reflected waves, with a reflection coefficient equal to 1.
In Region II, the wavefunction decays exponentially as an evanescent wave, penetrating a finite distance into the classically forbidden region.
The penetration depth is given by the skin depth parameter derived from the imaginary wave vector.
Boundary Conditions on Wavefunctions
The wavefunction ψ(x) must be continuous everywhere, so ψI(0) = ψII(0).
The first derivative of the wavefunction dψ/dx must be continuous at x = 0 because the potential is finite (not an infinite delta function).
Continuity of the derivative ensures probability current density is conserved across the boundary.
Derivations
Reflection and Transmission Coefficients for E > V<sub>0</sub>
Conditions: Particle energy E is strictly greater than the step height V0
Write Schrödinger equations for Region I: d2ψI/dx2 + k12ψI = 0, where k1 = \frac{\sqrt{2mE}}{\hbar}.
Write Schrödinger equation for Region II: d2ψII/dx2 + k22ψII = 0, where k2 = \frac{\sqrt{2m(E - V_0)}}{\hbar}.
Express general solutions as ψI(x) = Ae^{ik_1x} + Be^{-ik_1x} and ψII(x) = Ce^{ik_2x}.
Apply boundary conditions at $x = 0: A + B = C$ and k1(A - B) = k2C.
Solve for amplitude ratios B/A (reflection) and C/A (transmission), then calculate $R = |B/A|^{2}$ and $T = \frac{k_2}{k_1}|C/A|^{2}$.
Transmission Coefficient (T) for E > V<sub>0</sub>
$$T = \frac{4k_1 k_2}{(k_1 + k_2)^2}$$
T = probability of transmission Dimensionless
E > V0
Probability Current Conservation
$$R + T = 1$$
Sum of reflected and transmitted probability currents Dimensionless
Steady-state condition
Decay constant in Region II (E < V<sub>0</sub>)
$$\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$$
$\alpha = spatial decay rate$ m-1
E < V0
Reflection Coefficient for E < V<sub>0</sub>
$$R = 1$$
Total internal reflection equivalent in quantum mechanics Dimensionless
E < V0
Schrodinger Equation in Region II
$$-\frac{\hbar^2}{2m} \frac{d^2\psi_{II}}{dx^2} + V_0 \psi_{II} = E \psi_{II}$$
ψII = wavefunction in step region Joule (energy equation)
x ≥ 0
Graphs and experiments
Potential Step Function V(x)
Draw: A step graph with V = 0 for x < 0 and a vertical jump to $V = V_{0} at x = 0$, remaining constant for x > 0.
Interpret: Represents a sudden change in potential energy encountered by an incident particle.
Exam use: Directly tested in TGT/PGT matching questions and conceptual definition layout.
Probability Density for E > V<sub>0</sub>
Draw: An oscillating interference pattern in Region I due to incident and reflected waves, transitioning to a smaller amplitude constant oscillation in Region II.
Interpret: Shows partial reflection and partial transmission across the boundary.
Exam use: Used to explain nodes, antinodes, and amplitude changes at boundaries.
Wavefunction Decay for E < V<sub>0</sub>
Draw: A sinusoidal wave in Region I reflecting with a phase shift, and an exponential decay curve ψ(ψ)* dropping smoothly in Region II.
Interpret: Demonstrates quantum penetration into the classically forbidden region.
Exam use: Asked in PGT questions regarding evanescent waves and skin depth.
Solved numericals
TGT core
An electron with total energy E approaches a step potential of height V0 = 2 eV. If the kinetic energy in Region I is 4 eV, determine the ratio of wave vectors k2/k1.
Identify given values: $E = 4 + 2 = 6 eV$, V0 = 2 eV. Kinetic energy in Region II is E - V0 = 6 - 2 = 4 eV. Wait, let's use direct energy values: given total energy E = 4 eV, V0 = 2 eV.
Calculate kinetic energy in Region I = E = 4 eV. Kinetic energy in Region I$I = E - V_{0} = 4 - 2 = 2 eV$.
Use wave vector formula: k1 ∝ \sqrt{E} and k2 ∝ \sqrt{E - V_0}.
The potential step is assumed to be infinitely sharp (discontinuous jump at x = 0), which is an idealized mathematical model.
The particle is treated as non-relativistic, limiting validity to velocities much less than the speed of light.
Common misconceptions
Common error: Particles with energy greater than the potential step never reflect because they possess enough energy to clear the step.
Correct: Quantum mechanically, partial reflection occurs whenever there is an impedance mismatch in wave vector space, even if E > V0.
Wave properties dictate that boundaries cause partial reflection due to changes in the wavelength across the discontinuity.
Common error: When E < V0, the particle cannot enter the region x ≥ 0 at all.
Correct: The wavefunction penetrates into the classically forbidden region as an evanescent wave with an exponential decay.
Probability density is non-zero within a finite skin depth inside the barrier.
Quick revision
For E > V0, both reflection and transmission occur with coefficients R + T = 1.
For E < V0, total reflection takes place (R = 1, T = 0) accompanied by exponential decay in the forbidden region.
Wave vector in Region I: $k_1 = \frac{\sqrt{2mE}}{\hbar}$
Must remember
For a step potential defined as $V(x) = 0$ for $x < 0$ and $V(x) = V_0$ for $x \ge 0$, space is divided into two distinct regions.
When particle energy $E > V_0$, the reflection coefficient is given by $R = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2$ and transmission coefficient by $T = \frac{4k_1 k_2}{(k_1 + k_2)^2}$.
When particle energy $E < V_0$, total internal reflection occurs, yielding a reflection coefficient $R = 1$ and an exponentially decaying evanescent wave in Region II.
Boundary conditions require both the wavefunction $\psi(x)$ and its first derivative $\frac{d\psi}{dx}$ to be continuous at $x = 0$ due to a finite potential step.
Probability current conservation is strictly maintained, such that $R + T = 1$ for $E > V_0$.
Glossary
Step PotentialA potential energy function that changes discontinuously from $0$ to $V_0$ at a specific boundary $x = 0$.
Wave Vector ($k_1, k_2$)Spatial frequency parameters representing the momentum states of the quantum particle in respective regions, defined as $\frac{\sqrt{2mE}}{\hbar}$ and $\frac{\sqrt{2m(E - V_0)}}{\hbar}$.
Reflection Coefficient ($R$)The dimensionless probability ratio of the reflected particle probability current to the incident particle probability current.
Transmission Coefficient ($T$)The dimensionless probability ratio of the transmitted particle probability current to the incident particle probability current.
Evanescent WaveA non-propagating wavefunction that decays exponentially into a classically forbidden region where $E < V_0$.
Decay Constant ($\alpha$)The spatial decay rate parameter in the classically forbidden region, given by $\frac{\sqrt{2m(V_0 - E)}}{\hbar}$.
Both classical and quantum predict no permanent transmission, but quantum allows finite penetration depth
Practice Quiz
Choose an option to reveal the explanation.
1. What is the mathematical definition of a one-dimensional step potential $V(x)$?
Explanation: A standard step potential is defined as $V(x) = 0$ for $x < 0$ (Region I) and $V(x) = V_0$ for $x \ge 0$ (Region II).
2. For a particle with energy $E > V_0$ encountering a step potential, what does quantum mechanics predict about its reflection?
Explanation: Unlike classical mechanics, quantum mechanics dictates that whenever there is a change in the wave vector across a boundary, partial reflection occurs even if $E > V_0$.
3. What is the expression for the wave vector $k_1$ in Region I ($x < 0$) for a particle of mass $m$ and energy $E$?
Explanation: From the Schrödinger equation in Region I where $V=0$, the wave vector is derived as $k_1 = \frac{\sqrt{2mE}}{\hbar}$.
4. In Region II ($x \ge 0$) when $E > V_0$, what is the wave vector $k_2$?
Explanation: The effective kinetic energy in Region II is $E - V_0$, making the wave vector $k_2 = \frac{\sqrt{2m(E - V_0)}}{\hbar}$.
5. What is the formula for the reflection coefficient $R$ when $E > V_0$?
Explanation: By matching boundary conditions for amplitudes, the reflection coefficient for $E > V_0$ evaluates to $R = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2$.
6. What is the sum of the reflection coefficient $R$ and transmission coefficient $T$ for a steady-state particle beam at a step potential ($E > V_0$)?
Explanation: Probability current conservation requires that the sum of the reflected and transmitted probability current densities equals the incident current density, leading to $R + T = 1$.
7. When the particle energy is less than the potential step ($E < V_0$), what is the reflection coefficient $R$?
Explanation: For $E < V_0$, total internal reflection occurs, meaning the particle is completely reflected, resulting in $R = 1$.
8. How does the wavefunction behave in Region II ($x \ge 0$) when $E < V_0$?
Explanation: In the classically forbidden region ($E < V_0$), the solution to the Schrödinger equation is an exponentially decaying evanescent wave.
9. What is the decay constant $\alpha$ in Region II when $E < V_0$?
Explanation: The spatial decay rate parameter (decay constant) for $E < V_0$ is given by $\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar}$.
10. Why must the first derivative of the wavefunction $\frac{d\psi}{dx}$ be continuous at the boundary $x = 0$?
Explanation: Since the potential is finite (not a Dirac delta function), the Schrödinger equation requires that $\frac{d^2\psi}{dx^2}$ remains finite, implying continuous $\frac{d\psi}{dx}$.
11. What is the Schrödinger equation in Region II ($x \ge 0$) for a step potential?
Explanation: The time-independent Schrödinger equation in Region II includes the potential energy term $V_0 \psi_{II}$, yielding $-\frac{\hbar^2}{2m} \frac{d^2\psi_{II}}{dx^2} + V_0 \psi_{II} = E \psi_{II}$.
12. Which statement correctly contrasts classical and quantum mechanics for a particle approaching a step potential with $E > V_0$?
Explanation: Classically, a particle simply slows down at the step without reflecting. Quantum mechanically, wave impedance mismatch results in a non-zero reflection probability.
13. What is the physical unit of the wave vectors $k_1$ and $k_2$?
Explanation: Wave vectors represent spatial frequencies with units of inverse meters (m$^{-1}$). wavefunction propagation constants.
14. In the case where $E < V_0$, what happens to the probability density of finding the particle deep inside Region II ($x \to \infty$)?
Explanation: Because the wavefunction decays exponentially as $\psi_{II}(x) \propto e^{-\alpha x}$, the probability density $|\psi_{II}(x)|^2$ drops to zero as $x \to \infty$.
15. What does the boundary condition $\psi_I(0) = \psi_{II}(0)$ signify physically?
Explanation: The continuity of the wavefunction $\psi(x)$ across $x = 0$ ensures that the probability density is continuous and single-valued.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A quantum particle of mass $m$ and energy $E$ encounters a step potential where $V(x) = 0$ for $x < 0$ and $V(x) = V_0$ for $x \ge 0$. If $E > V_0$, what is the transmission coefficient $T$ expressed in terms of $k_1$ and $k_2$?
Explanation: The transmission coefficient for $E > V_0$ derived from probability current matching is given by $T = \frac{4k_1 k_2}{(k_1 + k_2)^2}$.
2. For a particle with $E < V_0$ incident on a potential step at $x = 0$, what is the nature of the wavefunction in Region I ($x < 0$)?
Explanation: When $E < V_0$, total reflection occurs ($R=1$). The superposition of the incident and reflected waves of equal amplitude in Region I forms a standing wave.
3. In a one-dimensional step potential problem, if the kinetic energy in Region I is four times the potential step height $V_0$ ($E = 4V_0$), what is the ratio of wave vectors $\frac{k_2}{k_1}$?
4. Which of the following derivative conditions must hold true across the boundary at $x = 0$ for a finite step potential?
Explanation: Since the potential step is finite (not infinite), the first derivative of the wavefunction $\frac{d\psi}{dx}$ must be continuous across $x = 0$.
5. What is the physical significance of the penetration depth (skin depth) when $E < V_0$ in a step potential?
Explanation: The skin depth parameter derived from the decay constant $\alpha$ defines the characteristic length scale over which the evanescent wave penetrates into the classically forbidden region ($x \ge 0$).
6. If the potential step height $V_0$ approaches zero, what do the reflection and transmission coefficients for $E > V_0$ become?
Explanation: When $V_0 \to 0$, $k_2 \to k_1$, making $R = \left( \frac{k_1 - k_1}{k_1 + k_1} \right)^2 = 0$ and $T = \frac{4k_1^2}{(2k_1)^2} = 1$. There is no barrier, so full transmission occurs.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. For a particle encountering a step potential with $E > V_0$, under what condition will the reflection coefficient $R$ be exactly zero?
Explanation: Reflection occurs due to the potential discontinuity. If $V_0 = 0$, there is no step, so $k_1 = k_2$ and $R = 0$.
2. What is the behavior of the transmission coefficient $T$ as the particle energy $E$ becomes extremely large compared to $V_0$ ($E \gg V_0$)?
Explanation: When $E \gg V_0$, $k_2 \approx k_1$, leading to $R \to 0$ and $T \to 1$, matching classical expectations at high energies.
3. In the region $x < 0$ ($V=0$), the general solution for the wavefunction is $\psi_I(x) = A e^{i k_1 x} + B e^{-i k_1 x}$. What do coefficients $A$ and $B$ represent respectively?
Explanation: In Region I, $A e^{i k_1 x}$ represents the wave traveling in the positive $x$-direction (incident wave) and $B e^{-i k_1 x}$ represents the wave traveling in the negative $x$-direction (reflected wave).
4. What is the value of the reflection coefficient $R$ when $E = V_0$?
Explanation: At $E = V_0$, $k_2 = 0$, meaning the wave vector in Region II vanishes, resulting in total reflection ($R = 1$). Energy is insufficient to propagate past the step.
5. Which of the following describes the evanescent wave in Region II when $E < V_0$?
Explanation: To keep the wavefunction finite as $x \to \infty$, the physically admissible solution in Region II for $E < V_0$ is only the decaying exponential term $C e^{-\alpha x}$ (setting the growing exponential coefficient $D = 0$).
6. How does the probability current density $J$ behave across the boundary at $x = 0$ in a steady-state step potential problem?
Explanation: Probability current conservation requires that the incident current equals the sum of the reflected and transmitted currents, ensuring $J$ is continuous across $x = 0$.
7. If the mass of the particle $m$ increases while $E$ and $V_0$ remain constant ($E > V_0$), how does the reflection coefficient $R$ change?
Explanation: Both $k_1$ and $k_2$ contain $\sqrt{m}$ proportionally. When substituting into $R = \left( \frac{k_1 - k_2}{k_1 + k_2} \right)^2$, the mass term cancels out entirely, making $R$ independent of $m$.
8. What is the dimension of the decay constant $\alpha$ when $E < V_0$?
Explanation: Since $\alpha$ appears in the exponent as $-\alpha x$, the product $\alpha x$ must be dimensionless, making the dimension of $\alpha$ equal to m$^{-1}$.
9. In the context of wave mechanics at a potential step, why does classical mechanics fail for $E > V_0$?
Explanation: Classical mechanics treats particles strictly as point masses that slow down without reflecting when $E > V_0$, failing to account for wave reflection caused by impedance mismatches at boundaries.
10. What happens to the penetration depth (skin depth $\delta = \frac{1}{\alpha}$) as the particle energy $E$ approaches the potential step $V_0$ from below ($E \to V_0$)?
Explanation: As $E \to V_0$, the decay constant $\alpha = \frac{\sqrt{2m(V_0 - E)}}{\hbar} \to 0$, which causes the skin depth $\delta = \frac{1}{\alpha} \to \infty$, allowing deep penetration into the barrier.