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Chapter 2: Problem 16

Calculate the energies and wavefunctions for a particle in a one-dimensional square well in which the potential energy rises to a finite value \(V\) at each end, and is zero inside the well; that is \\[ \begin{array}{ll} V(x)=V & x \leq 0 \text { and } x \geq L \\ V(x)=0 & 0

Short Answer

Expert verified
The exercise demonstrates that for any potential \(V\) and well width \(L\), there is always at least one bound level. Furthermore, as \(V\) approaches infinity, the solutions coincide with a specific equation. The energies can be calculated graphically from the intersection points using the equation \(E = h^2 k^2 / (2m)\).

Step by step solution

01

Divide the Space Into Zones

The space can be divided into three zones: x \leq 0, 0 < x < L and x \geq L considering the potential energy \( V(x) = V \) for x \leq 0 and x \geq L and \( V(x) = 0 \) for 0 < x < L.
02

Solve the Schrödinger Equations

The Schrödinger equations can be solved in three regions: \n Region I (x<=0), \( -(\hbar^2/2m) (d^2 \psi / dx^2) + V \psi = E \psi \)\n Region II (0=L), \( -(\hbar^2/2m) (d^2 \psi / dx^2) + V \psi = E \psi \)
03

Apply Boundary Conditions

We impose the boundary conditions: the finiteness and continuity of \(\psi\) and continuity of \(\mathrm{d} \psi / \mathrm{d} x\) across the zone boundaries.
04

Combine into a Single Equation

After imposing boundary conditions and some mathematical manipulations, we arrive at the following equation: \( k L+2 \arcsin \left\{\frac{k \hbar}{(2 m V)^{1 / 2}}\right\}=n \pi \quad k \hbar=(2 m E)^{1 / 2} \).
05

Graphical Solution

Plot the left and right side of the equation against \(k\). The intersection points will give the allowed values of \(k\). From these \(k\) values, calculate the energies using the equation \(E = h^2 k^2 / (2m)\).

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Most popular questions from this chapter

A very simple model of a polyene is the free electron molecular orbital (FEMO) model. Regard a chain of \(N\) conjugated carbon atoms, bond length \(R_{\mathrm{CC}},\) as forming a box of length \(L=(N-1) R_{\mathrm{CC}} .\) Find an expression for the allowed energies. Suppose that the electrons enter the states in pairs so that the lowest \(\frac{1}{2} N\) states are occupied. Estimate the wavelength of the lowest energy transition, taking \(R_{\mathrm{CC}}=140 \mathrm{pm}\) and \(N=22 .\) Repeat the calculation of the wavelength if the length of the chain is taken to be \((N+1) R_{\mathrm{CC}}(\) an assumption that allows for electrons to spill over the ends slightly.

Consider a harmonic oscillator of mass \(m\) undergoing harmonic motion in two dimensions \(x\) and \(y .\) The potential energy is given by \(V(x, y)=\frac{1}{2} k_{x} x^{2}+\frac{1}{2} k_{y} y^{2}\). (a) Write down the expression for the hamiltonian operator for such a system. (b) What is the general expression for the allowable energy levels of the two- dimensional harmonic oscillator? (c) What is the energy of the ground state (the lowest energy state)? Hint. The hamiltonian operator can be written as a sum of operators.

For a particle in a box, the mean value and mean square value of the linear momentum are given by \(\int_{0}^{L} \psi^{*} p \psi \mathrm{d} x\) and \(\int_{0}^{L} \psi^{*} p^{2} \psi \mathrm{d} x,\) respectively. Evaluate these quantities. Form the root mean square deviation \(\Delta p=\left\\{\left\langle p^{2}\right\rangle-\langle p\rangle^{2}\right\\}^{1 / 2}\) and investigate the consistency of the outcome with the uncertainty principle. Hint. Use \(p=(\hbar / \mathrm{i}) \mathrm{d} / \mathrm{d} x .\) For \(\left\langle p^{2}\right\rangle\) notice that \(E=p^{2} / 2 m\) and we already know \(E\) for each \(n\). For the last part, form \(\Delta x \Delta p\) and show that \(\Delta x \Delta p \geq \frac{1}{2} \hbar,\) the precise form of the principle, for all \(n\) evaluate \(\Delta x \Delta p\) for \(n=1.\)

The root mean square deviation of the particle from its mean position is \(\Delta x=\left\\{\left\langle x^{2}\right\rangle-\langle x\rangle^{2}\right\\}^{1 / 2} .\) Evaluate this quantity for a particle in a well and show that it approaches its classical value as \(n \rightarrow \infty\). Hint. Evaluate \(\left\langle x^{2}\right\rangle=\int_{0}^{L} x^{2} \psi^{2}(x) \mathrm{d} x\) In the classical case the distribution is uniform across the box, and so in effect \(\psi(x)=1 / L^{1 / 2}.\)

The oscillation of the atoms around their equilibrium positions in the molecule HI can be modelled as a harmonic oscillator of mass \(m \approx m_{\mathrm{H}}\) (the iodine atom is almost stationary \()\) and force constant \(k_{\mathrm{f}}=313.8 \mathrm{N} \mathrm{m}^{-1} .\) Evaluate the separation of the energy levels and predict the wavelength of the light needed to induce a transition between neighbouring levels.
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