Scanning Tunnelling Microscope

Figure 20.9:

The scanning tunnelling microscope (STM) for which a schematic diagram is shown in Figure 20.9 uses quantum tunnelling for its functioning. A very narrow tip usually made of tungsten or gold and of the size of the order of $1 \, \AA$ or less is given a negative bias voltage. The tip scans the surface of the sample which is given a positive bias. The tip is maintained at a small distance from the surface as shown in the figure. Figure 20.10 shows the potential experienced by an electron respectively in the sample, tip and the vacuum in the gap between the sample and the tip. As the tip has a negative bias, and electron in the tip is at a higher potential than in the sample. As a consequence the electrons will flow from the tip to the sample setting up a current in the circuit. This is provided the electrons can tunnel through the potential barrier separating the tip and the sample. The current in the circuit is proportional to the tunnelling transmission coefficient $T$ calculated earlier. This is extremely sensitive to the size of the gap $a$.

Figure 20.10:

In the STM the tip is moved across the surface of the sample. The current in the circuit differs when the tip is placed over different points on the sample. The tip is moved vertically so that the current remains constant as it scans across the sample. This vertical displacement recorded at different points on the sample gives an image of the surface at the atomic level. Figure 20.11 shows an STM image of a graphite sample.

Figure 20.11:

Problems

1. Consider a particle at time $t=0$ with the wave function
   
   $$\psi(x,t=0)=A \exp[-(x-a)^2/(4 L^2)]$$
   
   
   and $L$ are constants of dimension length. Given $\int_{-\infty}^{\infty} \exp[-x^2/2] \, dx = \sqrt{2 \pi}$.

   a.

   Determine the normalization constant A.

   b.

   What is the expectation value $\langle x \rangle$?

   c.

   How much is $\Delta x$, the uncertainty in $x$?

   d.

   What is the momentum expectation value $\langle p \rangle$?

   e.

   How much is $\Delta p$, the uncertainty in $p$?

   f.

   What happens to the uncertainty in $x$ and the uncertainty $p$ if $L$ is increased?

   d.

   How does the product $\Delta x \, \Delta p$ change if $L$ is varied?

2. A free particle of mass $m$ and energy $E$ is incident on a step potential barrier $V$. What is the wave function $\psi(x,t)$ of the particle inside the step potential if (a.) (b.) $V>E$.
3. A particle of mass $m$ is confined to $0\le x \le L$ by two very high step potentials (particle in a box).

   a.

   What is the wave function $\psi_1(x,t)$ of the particle in the lowest energy state?

   b.

   What is the wave function $\psi_2(x,t)$ of the particle in the first excited energy state?

   c.

   For $\psi_1(x,t)$, what are the expectation values $\langle x \rangle$ and $\langle p \rangle$?

   d.

   For $\psi_1(x,t)$, what are the uncertainties $\Delta x$ and $\Delta p$.

4. An electron trapped in a region of length $L=\, 10 \, \AA$. (a.)What are the energies of the ground state and the first two excited states? (b.) An electron in the first excited state emits radiation and de-excites to the ground state. What is the wavelength of the emitted radiation?
5. The wave function of a particle confined in a region of length $L$ is given to be
   
   $$\psi(x,t)=\frac{3}{5} \psi_1(x,t)+\frac{4}{5} \psi_2(x,t)$$
   
   
   where $\psi_1(x,t)$ and $\psi_2(x,t)$ are the wave functions introduced in Problem 6. What are the possible outcomes and their probabilities if the energy of the particle is measured? What is the expectation value of the energy? What is the uncertainty in the energy?

   Figure 20.12:

6. A particle of mass $m$ and energy $E$ is incident from zero potential to a step potential $V$, where $V=2 E$ as shown in Figure 20.12.

   The incident, reflected and transmitted wave functions $\psi_I$, $\psi_R$ and respectively are
   

   $$\psi_I(x,t)=A e^{-i (E t - p x)/\hbar}\,,\hspace{1cm} \psi_R(x,t)=B e^{-i (E t + p x)/\hbar}\hspace{0.5cm}$$
   
   
   
   
   $${\rm and} \hspace{0.5cm} \psi_T(x,t)=C e^{- (i E t + a x)/\hbar}\,.$$
   
   
   where $p$ and $a$ are constants.

   a.

   What is the ratio ?

   b.

   Match the boundary conditions at $x=0$ to determine the ratio $(A+B)/C$.

   c.

   Match the boundary conditions at $x=0$ to determine the ratio $(A-B)/C$.