Chapter 6, Sectionally Constant Potentials in One Dimension Video Solutions, Quantum mechanics

Problem 1

Obtain the transmission coefficient for a rectangular potential barrier of width $2 a$ if the energy exceeds the height $V_0$ of the barrier. Plot the transmission coefficient as a function of $E / V_0$ (up to $E / V_0=3$ ), choosing $\left(2 m a^2 V_0\right)^{1 / 2}=(3 \pi / 2) \hbar$.

Gopesh Vishwakarma

Numerade Educator

03:21

Problem 2

Consider a potential $V=0$ for $x>a, V=-V_0$ for $a \geq x \geq 0$, and $V=+\infty$ for $x<0$. Show that for $x>a$ the positive energy solutions of the Schrödinger equation have the form
$$
e^{i(k x+2 \delta)}-e^{-i k x}
$$

Calculate the scattering coefficient $\left|1-e^{2 i \delta}\right|^2$ and show that it exhibits maxima (resonances) at certain discrete energies if the potential is sufficiently deep and broad.

Manish Jain

Numerade Educator

Problem 3

A particle of mass $m$ moves in the one-dimensional double well potential
$$
V(x)=-g \delta(x-a)-g \delta(x+a)
$$

If $g>0$, obtain transcendental equations for the bound-state energy eigenvalues of the system. Compute and plot the energy levels in units of $\hbar^2 / m a^2$ as a function of the dimensionless parameter $m a g / \hbar^2$. Explain the features of this plot. In the limi of large separation, $2 a$, between the wells, obtain a simple formula for the splittin $\Delta E$ between the ground state (even parity) energy level, $E_{+}$, and the excited (odc parity) energy level, $E_{-}$.

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Problem 4

Problem 3 provides a primitive model for a one-electron linear diatomic molecule with interatomic distance $2 a=|X|$, if the potential energy of the "molecule" is taken as $E_{ \pm}(|X|)$, supplemented by a repulsive interaction $\lambda g /|X|$ between the wells ("atoms"). Show that, for a sufficiently small value of $\lambda$, the system ("molecule") is stable if the particle ("electron") is in the even parity state. Sketch the total potential energy of the system as a function of $|X|$.

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Problem 5

If the potential in Problem 3 has $g<0$ (double barrier), calculate the transmission coefficient and show that it exhibits resonances. (Note the analogy between the system and the Fabry-Perot étalon in optics.)

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13:19

Problem 6

. A particle moves in one dimension with energy $E$ in the field of a potential defined as the sum of a Heaviside step function and a delta function:
$$
V(x)=V_0 \eta(x)+g \delta(x) \quad \text { (with } V_0 \text { and } g>0 \text { ) }
$$

The particle is assumed to have energy $E>V_0$.
(a) Work out the matrix $M$, which relates the amplitudes of the incident and reflected plane waves on the left of the origin $(x<0)$ to the amplitudes on the right $(x>0)$.
(b) Derive the elements of the matrix $S$, which relates incoming and outgoing amplitudes.
(c) Show that the $S$ matrix is unitary and that the elements of the $S$ matrix satisfy the properties expected from the applicable symmetry considerations.
(d) Calculate the transmission coefficients for particles incident from the right and for particles incident from the left, which have the same energy (but different velocities).

Daniel Sneed

Numerade Educator

02:00

Problem 7

For the potentials in Problems 5 and 6, verify the identity
$$
|r|^2+|t|^2=\operatorname{Im} t
$$
for the complex-valued amplitudes $r$ and $t$, if the elements of the $S$ matrix are expressed as $S_{11}=2 i r$ and $S_{21}=1+2 i t$.

Ajay Singhal

Numerade Educator