Characteristics of Quantum Wavefunctions

Chem 3240 · Lecture 3.5

Davit Potoyan

Quantum objects go where classical ones cannot

  • Tunneling into classically forbidden regions
  • Electron emission, ammonia inversion, tunnel diodes
  • The finite well is the realistic model that allows it

Reading a wavefunction qualitatively

  • Curvature is set by the sign of \(E - V\)
  • \(E > V\): \(\psi\) oscillates, curves toward axis
  • \(E < V\): \(\psi\) decays, curves away from axis

Curvature follows the potential

  • A stepped potential changes the local wavelength
  • Higher kinetic energy means shorter wavelength
  • \(\psi\) stays smooth and continuous everywhere

Three regions, three solutions

The finite square well: \(V=0\) inside, \(V=V_0\) outside.

\[ -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{d{x}^2}+V(x)\psi(x) =E\psi(x) \]

\[ \begin{gathered} \text{Region I: } V(x) = V_0 ~~~~~~~~~~~~~~~~ x \leq-\frac{L}{2} ~~~~~ \psi_{I} = A e^{\beta x} \\ \text{Region II: } V(x) = 0 ~~~ -\frac{L}{2} \leq x \leq\frac{L}{2} ~~~~~~~ \psi_{II} = B \cos(k x) + C \sin(k x)\\ \text{Region III:} V(x) = V_0 ~~~~~~~~~~~~~~ x \geq \frac{L}{2} ~~~~~~~ \psi_{III} = D e^{-\beta x} \end{gathered} \]

Inside vs outside the well

Inside (\(E > V\)) \[ \psi(x)=A\sin(kx)+B\cos(kx) \] ::: {.fragment} Oscillatory.

Outside (\(E < V_0\)) ::: {.fragment} \[ k^2 = \frac{2m(V_0 - E)}{\hbar^2} \]

Real \(\beta\): exponential decay, an evanescent wave.

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Why continuity matters

  • Match \(\psi\) and \(\psi'\) at each boundary

\[ \psi_{I}\!\left(-\tfrac{L}{2}\right) = \psi_{II}\!\left(-\tfrac{L}{2}\right) \]

  • \(\psi'\) continuous so \(\psi''\) exists
  • \(\psi''\) is the kinetic energy operator: must be defined

Even and odd solutions

The well is symmetric about \(x=0\), so states split by parity.

Even (\(\psi(x)=\psi(-x)\)) \[ \psi_{II}(x) = B \cos(k x) \] \[ \beta = k\tan\left(k\frac{L}{2}\right) \]

Odd (\(\psi(x)=-\psi(-x)\)) \[ \psi_{II}(x) = C \sin(k x) \] \[ \beta = -\frac{k}{\tan\left(k\frac{L}{2}\right)} \]

No closed form: solve graphically or numerically.

Wavefunctions leak out

  • \(\psi\) extends into the classically forbidden region
  • Green area: probability of being outside the well
  • Leakage grows as \(V_0 - E \to 0\)

Tunneling probability

The chance of finding the particle outside the box:

\[ P\left(-\frac{L}{2}>x>\frac{L}{2}\right)=\frac{\int^{\frac{-L}{2}}_{-\infty} |\psi(x)|^2\ dx +\int^{+\infty}_{\frac{L}{2}} |\psi(x)|^2\ dx }{\int_{-\infty}^{+\infty} |\psi(x)|^2\ dx } \]

  • States near the top of the well tunnel most

Finite vs infinite well

  • Finite well has a finite number of bound states
  • Low states: energies nearly match the infinite well
  • High states: \(\psi\) spreads out, longer wavelength
  • Longer wavelength means smaller \(k\), so lower energy

Takeaway

Wavefunctions stay smooth, curving and decaying according to \(E - V\), and they leak into forbidden regions, giving the finite well its tunneling and its limited set of bound states.