Quantum Mechanics Notes

Differentiability of Wave Functions

• The solutions for energy eigenstates violate the condition that the wave function must be differentiable at all values of $x$.
• For example, $\sqrt{\frac{2}{l}} \sin(\frac{3 \pi x}{l})$ is a solution, but its graph shows it's piecewise continuous, not differentiable at $x = 0$ because the slope changes abruptly.
• The slope of the wave function has physical meaning, and the square of the wave function is the probability density.

Addressing the Differentiability Issue

• The non-differentiability is not a serious concern because the infinite potential used is unrealistic; a finite potential is more physical.
• With a finite potential (large but finite) for $x < 0$ and $x > l$, the functions smooth out.
• The function exhibits exponential decay to the left of $x = 0$ and to the right of $x = l$.
• The length of this exponential decay is related to the size of the potential; the higher the potential, the smaller the width of the decay.

Particle in a Finite Well

• The particle in a finite well problem involves a potential function with a finite height $V_0$.
• Different potential functions lead to different problems in quantum mechanics.
• The solutions have a slightly different mathematical form but still represent the shape of the wave function.
• For instance, if $x = 0$ is at the center of the well, the solution is a cosine function, not a sine function.

Energy Levels in the Finite Well

• Energy levels: Below $V<em>0$, discrete energy eigenvalues (e.g., $E</em>1$, $E_2$) exist.
• Above $V_0$, the energy levels become continuous, resembling free particle problems.
• Infinite well: Only certain energy levels are allowed (e.g., $E<em>2 = 4E</em>1$, $E<em>3 = 9E</em>1$).
• Distinction: Discrete (quantized) vs. continuous energy values.

Classical vs. Quantum Connection

• If the energy level is below $V_0$, classically, the particle is trapped, leading to discrete energy values.
• If the energy level is above $V_0$, classically, the particle can escape, behaving more like a free particle.
• There are connections between classical and quantum mechanics; the solutions have similarities.

Bound vs. Unbound States

• Bound states: When a particle cannot escape to infinity. If it's in one of these energy eigenstates, it will never be found at $\pm \infty$ .
• Unbound states: The particle is not confined and can exist at infinity.

Mathematical Formulation of the Finite Well Problem

• Solutions are sought in the region between $\frac{-l}{2}$ and $\frac{+l}{2}$, similar to the infinite well problem.
• In this region, $V = 0$, and the solutions are the same as before.
• The general solution involves sine and cosine functions:
  • $c<em>1 \sin(k</em>e x)$
  • $c<em>2 \cos(k</em>e x)$
  • $c<em>3 e^{i k</em>e x}$
  • $c<em>4 e^{-i k</em>e x}$
• $k_e = \sqrt{\frac{2mE}{\hbar^2}}$, which has units of wave number.

Solutions Outside the Well

• For x > \frac{l}{2} or x < \frac{-l}{2}, the differential equation is different because $V \neq 0$.
• The differential equation becomes:
  $-\frac{\hbar^2}{2m} \frac{d^2 \phi(x)}{dx^2} + V_0 \phi(x) = E \phi(x)$
• If E < V0, the term $E - V</em>0$ is negative.

Introducing Alpha

• Define $\alpha^2 = \frac{2m(V_0 - E)}{\hbar^2}$.
• The solutions outside the well are:
  • $c_3 e^{-\alpha x}$
  • $c_4 e^{\alpha x}$

Physical Considerations for Solutions

• In the region x > \frac{l}{2}, the solution must be a decaying exponential function:
  $c_3 e^{-\alpha x}$
• The solution $e^{\alpha x}$ is not physically realistic because it increases without bound as $x$ goes to infinity.

Lowest Energy Solution

• The lowest energy solution involves the cosine function and has exponential tails on either side.
• In the limit that $V_0$ goes to infinity, the solution for the finite well problem converges to the solution for the infinite well problem.

Characteristic Length

• $\alpha$ determines the characteristic length over which the exponential decays.
• Analogy to capacitor discharge, where $\tau$ is the time for the charge to decrease by 63%.
• The characteristic length is $\frac{1}{\alpha}$, with units in meters.
• We can write the solution form as $e^{\frac{-x}{1/\alpha}}$, where $\frac{1}{\alpha}$ is the length at which the function decays by 63%.

Finding Discrete Energy Values

• In the infinite well, energy values were restricted because the sine functions had to be zero at the boundaries.
• In the finite well, the functions must meet certain conditions at the boundaries.
• The functions must be continuous and differentiable.

Boundary Conditions

• At $x = \frac{l}{2}$, the solution inside and outside the well must be equal:
  $\phi<em>{inside}(x = \frac{l}{2}) = \phi</em>{outside}(x = \frac{l}{2})$
• The derivatives of the solutions inside and outside the well must also be equal:
  $\frac{d}{dx} \phi<em>{inside}(x = \frac{l}{2}) = \frac{d}{dx} \phi</em>{outside}(x = \frac{l}{2})$

Equations and Unknowns

• These conditions yield four equations and four unknowns.
• Two equations come from the continuity and differentiability at $x = \frac{l}{2}$.
• Two more equations come from similar conditions at $x = \frac{-l}{2}$.
• The constants $c<em>1$, $c</em>2$, $c<em>3$, and $c</em>4$ are the unknowns.
• If the first solution doesn't have a zero at the origin, then $c_2$ is not zero.