Quantum Wave Mechanics: Free Space Propagation and Bound States in Infinite Potential Wells

Administrative and Organizational Logistics

• Homework Assignment Schedule:
  • The next homework assignment is released today.
  • Next Monday is a designated holiday; no lecture will take place.
  • The default submission deadline for the homework is set for Tuesday evening.
  • Potential deadline extension: The deadline may shift to Wednesday evening if necessary, but students should plan around the Tuesday submission target.
  • Homework scheduling follows a Sunday/Monday periodicity cadence.

Non-Relativistic Schrödinger Equation and Wave Mechanics Review

• Space-Form Non-Relativistic Schrödinger Equation (Time-Independent):

  • $-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)$

• Time-Dependent Wave Function Component:

  • Features a sinusoidal waveform represented in complex exponential form: $\phi(t) = e^{-j\omega t} = e^{-j\frac{E}{\hbar}t}$
  • Energy $E$ acts as the separation constant arising from spatial and temporal separation of variables.
  • Angular frequency $\omega$ connects directly to energy through $E = \hbar\omega$.

• Probability Density Function and Fundamental Boundary Conditions:

  • The overall spatial-temporal probability density is defined as: $P(x,t) = \Psi(x,t)\Psi^*(x,t) = |\Psi(x,t)|^2$
  • Spatial probability density (time-independent): $P(x) = \psi(x)\psi^*(x) = |\psi(x)|^2$
  • Existence / Normalization Boundary Condition: The particle must exist somewhere in the total spatial domain, requiring the spatial integral over all space to equal unity: $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$
  • Continuity Conditions: The wave function $\psi(x)$ and its first derivative $\frac{d\psi(x)}{dx}$ must be continuous, finite, and single-valued across space for finite potentials $V(x)$.
  • These physical constraints align strictly with the Heisenberg Uncertainty Principle.

Free Particle Kinematics and Traveling Wave Solutions

• Free Space Scenario Definition:

  • Assumes an electron moving in free space without influence from atomic nuclei or periodic lattice potentials ($V(x) = C = 0$ everywhere).

• Simplified Differential Equation for Free Space ($V = 0$):

  • $-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} = E\psi(x)$
  • Rearranged standard second-order homogeneous form: $\frac{d^2\psi(x)}{dx^2} + \frac{2mE}{\hbar^2}\psi(x) = 0$

• Exponential Solution Form:

  • $\psi(x) = A e^{j \sqrt{\frac{2mE}{\hbar^2}} x} + B e^{-j \sqrt{\frac{2mE}{\hbar^2}} x}$

• Wave Number $k$ Definition:

  • $k = \sqrt{\frac{2mE}{\hbar^2}} = \frac{\sqrt{2mE}}{\hbar}$
  • $E = \frac{\hbar^2 k^2}{2m}$
  • Parabolic Dispersion Relation: Energy $E$ is proportional to $k^2$ ($E \propto k^2$), yielding a parabolic energy-wavevector curve.

• Total Spatial-Temporal Wave Function $\Psi(x,t)$:

  • $\Psi(x,t) = \psi(x) e^{-j\omega t} = A e^{j(kx - \omega t)} + B e^{-j(kx + \omega t)}$
  • Component $A e^{j(kx - \omega t)}$: Represents a plane wave traveling in the positive $x$-direction ($+x$).
  • Component $B e^{-j(kx + \omega t)}$: Represents a plane wave traveling in the negative $x$-direction ($-x$).

• Momentum and De Broglie Relationships:

  • Momentum magnitude: $p = \hbar k$
  • De Broglie wavelength: \lambda = \frac{h}{p} = \frac{2\pi\hbar}{p} = \frac{2\pi}{k}
  • Wave number in terms of wavelength: $k = \frac{2\pi}{\lambda}$

• Unidirectional Motion ($+x$ Direction Only, $B = 0$):

  • Wave function: $\Psi(x,t) = A e^{j(kx - \omega t)}$
  • Probability Density Evaluation:
  • $|\Psi(x,t)|^2 = \Psi(x,t)\Psi^<em>(x,t) = \left(A e^{j(kx - \omega t)}\right) \left(A^</em> e^{-j(kx - \omega t)}\right) = |A|^2$
  • Implication: The spatial dependence completely cancels out, making the probability density $|A|^2$ a constant across all positions $x$.
  • Heisenberg Uncertainty Relation: Having an exact, single wave vector $k$ implies precise momentum ($p = \hbar k$, so $\Delta p = 0$). Consequently, the spatial position uncertainty becomes infinite ($\Delta x \rightarrow \infty$), meaning the electron can be found anywhere in space with equal probability.
  • Localization: To construct a spatially localized electron (wave packet), one must combine a superposition of wave functions possessing a range of distinct $k$ values.

Bound Electrons: Infinite Potential Well Formulation

• Physical Motivation:

  • Electrons in solid-state devices, crystals, and semiconductors are rarely free; they are confined by localized potential wells $V(x)$.

• One-Dimensional Infinite Potential Well System Geometry:

  • Region $0 \le x \le a$ (Inside Well): $V(x) = 0$, width of well is $a$.
  • Region $x < 0$ and $x > a$ (Barriers): $V(x) = V_0 \rightarrow \infty$.

• Wave Function Behavior at Infinite Barriers ($V \rightarrow \infty$):

  • For any finite energy $E$, the potential energy inside the barrier is infinitely larger than $E$.
  • Wave function value outside the well: $\psi(x) = 0$ for $x < 0$ and $x > a$.
  • Boundary values: $\psi(0) = 0$ and $\psi(a) = 0$.
  • Exception to Derivative Continuity: The first derivative $\frac{d\psi(x)}{dx}$ is not continuous at the boundaries $x = 0$ and $x = a$ due to the non-physical infinite potential step $V \rightarrow \infty$.

• Internal Differential Equation ($0 \le x \le a$):

  • $\frac{d^2\psi(x)}{dx^2} + k^2\psi(x) = 0$, where $k = \sqrt{\frac{2mE}{\hbar^2}}$

• Trigonometric Solution Form:

  • $\psi(x) = A_1 \cos(kx) + A_2 \sin(kx)$

• Application of Boundary Conditions:

  • Boundary at $x = 0$:
  • $\psi(0) = A_1 \cos(0) + A_2 \sin(0) = A_1 (1) + A_2 (0) = A_1 = 0$
  • Since $\psi(0) = 0$, $A_1$ must equal zero, eliminating the cosine term entirely.
  • Remaining wave function: $\psi(x) = A_2 \sin(kx)$
  • Boundary at $x = a$:
  • $\psi(a) = A_2 \sin(ka) = 0$
  • To avoid trivial solutions ($A_2 = 0$, which implies no particle exists), we require $\sin(ka) = 0$.
  • Constraint equation: $ka = n\pi$, where $n$ is an integer.
  • Quantized wave vector values: $k_n = \frac{n\pi}{a}$ for $n = 1, 2, 3, \dots$
  • Note on quantum numbers: $n$ must be a positive integer ($n \in \mathbb{Z}^+$). Negative integers yield redundant states, and $n = 0$ results in $\psi(x) = 0$ everywhere.

• Normalization Determination of Coefficient $A_2$:

  • $\int_{0}^{a} |\psi(x)|^2 dx = \int_{0}^{a} A_2^2 \sin^2\left(\frac{n\pi x}{a}\right) dx = 1$
  • Using trigonometric identity $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$:
  • $A_2^2 \int_{0}^{a} \frac{1 - \cos\left(\frac{2n\pi x}{a}\right)}{2} dx = A_2^2 \left(\frac{a}{2}\right) = 1$
  • $A_2 = \sqrt{\frac{2}{a}}$

• Normalized Spatial Wave Function for State $n$:

  • $\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right)$

Energy Quantization, Standing Waves, and Physical Interpretations

• Quantized Energy Levels Derivation:

  • Equating $k_n = \frac{n\pi}{a}$ with $k_n = \sqrt{\frac{2mE_n}{\hbar^2}}$:
  • $k_n^2 = \frac{n^2 \pi^2}{a^2} = \frac{2mE_n}{\hbar^2}$
  • $E_n = \frac{n^2 \pi^2 \hbar^2}{2m a^2}$
  • Substitution with Planck's constant ($h = 2\pi\hbar$):
  • $E_n = \frac{n^2 h^2}{8m a^2}$

• Physical Differences Between Free and Bound States:

  • Free Electrons: Form continuous traveling waves $e^{j(kx - \omega t)}$.
  • Bound Electrons: Form localized standing waves $\sin\left(\frac{n\pi x}{a}\right)$.
  • Instrument Analogy: Equivalent to acoustic standing waves on guitar strings or inside woodwind instruments (clarients), where fixed physical boundaries restrict resonance to specific harmonic modes/frequencies.

• Ground State and Zero-Point Energy:

  • Lowest allowed quantum energy state occurs at $n = 1$:
  • $E_1 = \frac{\pi^2 \hbar^2}{2m a^2} > 0$
  • Key Principle: Ground state energy $E_1$ cannot be zero ($E_1 eq 0$) even though $V(x) = 0$ inside the well. A confined quantum particle cannot possess zero kinetic energy without violating spatial uncertainty constraints.

• Spatial Probability Density Distribution Characteristics $|\psi_n(x)|^2$:

  • Ground State ($n = 1$):
  • Features a single symmetric probability peak located directly in the center ($x = \frac{a}{2}$).
  • Probability decays smoothly towards zero at the hard boundary walls $x = 0$ and $x = a$.
  • Second State ($n = 2$):
  • Displays two symmetric probability peaks separated by an internal node (zero probability) at $x = \frac{a}{2}$.
  • Third State ($n = 3$):
  • Displays three probability peaks separated by two internal nodes.
  • High Quantum Numbers ($n \rightarrow \infty$) and Classical Correspondence:
  • As $n$ increases, spatial probability oscillations become exceedingly dense.
  • Averaging over macroscopic spatial intervals smooths the rapid oscillations into an effectively constant spatial line ($P \approx \frac{1}{a}$).
  • Demonstrates the Correspondence Principle: At high quantum numbers ($n \gg 1$), quantum mechanical predictions merge smoothly with classical macroscopic physics, where a classical bouncing particle has uniform spatial probability throughout a box.