Approximations

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Transmission or bound states

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Transmission or bound states

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Transmission or bound states

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Schrodinger equation for constant \(V(x) = V\)

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

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Schrodinger equation for constant \(V(x) = V\)

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

• Solution depends on value of \(E - V\):
• If energy is less than the potential \(E < V\), then we have evanescent waves:

\[ \psi(x) = A e^{- \kappa x} + B e^{\kappa x} \qquad \kappa^2 = \frac{2m}{\hbar ^2} (V - E) \]

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Schrodinger equation for constant \(V(x) = V\)

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

• Solution depends on value of \(E - V\):
• If energy is the same as the potential energy \(E = V\), then:

\[ \psi(x) = A + B \, x \]

Intro: transfer matrix method

• For 1D potential energy functions \(V(x)\) (here assume 1D systems)
• Approximation of potential energy \(V(x)\) by piece-wise constant \(V_i\)
• Schrodinger equation for constant \(V(x) = V\)

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

• Solution depends on value of \(E - V\):

Boundary conditions across a step in V(x)

• Suppose there is a step in the potential in \(x = a\).
• Boundary conditions: Continuity of wave function \(\psi(x)\) and derivative \(\frac{d\psi(x)}{dx}\):

\[ \begin{aligned} \psi_I(a) &= \psi_{II}(a) \qquad & A e^{i k_1 a} + B e^{-i k_1 a} &= C e^{i k_2 a} + D e^{-i k_2 a}\\ \frac{d\psi_I(a)}{dx} &= \frac{d\psi_{II}(a)}{dx} \qquad & i k_1 A e^{i k_1 a} -i k_1 B e^{-i k_1 a} &= i k_2 C e^{i k_2 a} -i k_2 D e^{-i k_2 a}\\ \end{aligned} \]

Boundary conditions across a step in V(x)

\[ \begin{aligned} \psi_I(a) &= \psi_{II}(a) \qquad & A e^{i k_1 a} + B e^{-i k_1 a} &= C e^{i k_2 a} + D e^{-i k_2 a}\\ \frac{d\psi_I(a)}{dx} &= \frac{d\psi_{II}(a)}{dx} \qquad & i k_1 A e^{i k_1 a} -i k_1 B e^{-i k_1 a} &= i k_2 C e^{i k_2 a} -i k_2 D e^{-i k_2 a}\\ \end{aligned} \]

\[ \pmatrix{1 & 1 \\ i k_1 & - i k_1} \pmatrix{e^{i k_1 a} & 0 \\ 0 & e^{-i k_1 a}} \pmatrix{A\\ B} = \pmatrix{1 & 1 \\ i k_2 & - i k_2} \pmatrix{e^{i k_2 a} & 0 \\ 0 & e^{-i k_2 a}} \pmatrix{C\\ D} \]

Boundary conditions across a step in V(x)

\[ \pmatrix{1 & 1 \\ i k_1 & - i k_1} \pmatrix{e^{i k_1 a} & 0 \\ 0 & e^{-i k_1 a}} \pmatrix{A\\ B} = \pmatrix{1 & 1 \\ i k_2 & - i k_2} \pmatrix{e^{i k_2 a} & 0 \\ 0 & e^{-i k_2 a}} \pmatrix{C\\ D} \]

Express coefficient \(A\), \(B\) in \(C\), \(D\):

\[ \pmatrix{A\\ B} = \pmatrix{e^{i k_1 a} & 0 \\ 0 & e^{-i k_1 a}}^{-1} \pmatrix{1 & 1 \\ i k_1 & - i k_1}^{-1} \\ \qquad \qquad \qquad \qquad \qquad \qquad \times\pmatrix{1 & 1 \\ i k_2 & - i k_2} \pmatrix{e^{i k_2 a} & 0 \\ 0 & e^{-i k_2 a}} \pmatrix{C\\ D} \]

Rename the matrices as function of \(V\) and \(a\):

\[ \pmatrix{A\\ B} = E_1^{-1}(a) K_1^{-1} K_2 E_2(a) \pmatrix{C\\ D} \]

Transfer matrix for a single step

\[ E_j(a) = \pmatrix{e^{i k_j a} & 0 \\ 0 & e^{-i k_j a}} , \qquad K_j \pmatrix{1 & 1 \\ i k_j & - i k_j} \]

\[ \pmatrix{A_1\\ B_1} = E_1^{-1}(a) K_1^{-1} K_2 E_2(a) \pmatrix{A_2\\ B_2} \]

We can define the transfer matrix for a single step:

\[ T_{12} = E_1^{-1}(a) K_1^{-1} K_2 E_2(a) \]

Connection between coefficient before/after step:

\[ \pmatrix{A_1\\ B_1} = T_{12} \, \pmatrix{A_2\\ B_2} \]

Multiple potential steps

Extending the relation over multiple steps:

\[ \pmatrix{A_1\\ B_1} = T_{12} \, \pmatrix{A_2\\ B_2} = T_{12} \, T_{23} \, \pmatrix{A_3\\ B_3} \]

In general, after N steps we obtain:

\[ \pmatrix{A_0\\ B_0} = T \pmatrix{A_{N+1}\\ B_{N+1}} = T_{01} \, T_{12} \, \dots \, T_{N,N+1} \pmatrix{A_{N+1}\\ B_{N+1}} \]

Or renaming the indices on the left and right:

\[ \pmatrix{A_L\\ B_L} = \pmatrix{t_{11} & t_{12}\\ t_{21} & t_{22}} \pmatrix{A_R\\ B_R} \]

Scattering and Bound states

Scattering: \(B_R = 0\)

\[ \pmatrix{A_L\\ B_L} = \pmatrix{t_{11} & t_{12}\\ t_{21} & t_{22}} \pmatrix{A_R\\ 0} \]

Therefore the transmission and reflection coefficients become:

\[ \begin{aligned} \textrm{Transmission}\quad & T(E) = |A_R/A_L|^2 = 1\,/\,|t_{11}(E)|^2\\ \textrm{Reflection}\quad & R(E) = |B_L/A_L|^2 = |t_{21}(E)|^2\,/\,|t_{11}(E)|^2\\ \end{aligned} \]

Scattering and Bound states

Bound states: \(A_L = 0, \quad B_R = 0\)

\[ \pmatrix{0\\ B_L} = \pmatrix{t_{11} & t_{12}\\ t_{21} & t_{22}} \pmatrix{A_R\\ 0} \Longrightarrow \begin{aligned} A_L = t_{11}(E) \,A_R\\ B_L = t_{21}(E) \,A_R\\ \end{aligned} \]

• Bound states are given by zeros of \(t_{11}\)
• The total wave function is defined upon the coefficients \(B_L\) and \(A_R\). We can obtain these unknowns by
  • first using the second equation: \(B_L = t_{21}(E) \,A_R\) to obtain \(B_L\), and then
  • applying normalization to the whole wave function to fix \(A_R\).