Question

de Broglie Matter waves and the non-relativistic electron: The Double slit experiment for electrons

(a) Electrons were accelerated through a potential difference of V = 50 kV. Is it permissible to use non- relativistic expressions to find the linear momentum of the electrons? Calculate the de Broglie wavelength of an electron in the Hitachi experiment

(b) The spacing between the fringes observed was 90 nm. [The image shown in Fig 1.4 in the textbook has been magnified further of course, to generate fringes on the imaging detector that were separated by about 1.4 mm.] Given the 90 nm fringe spacing, what is the spacing between the two slits?

(c) The distance to the screen was about 1.5 m, while the average spacing between two electrons was 150 km long, effectively ruling out the chances of two electrons being in the apparatus at the same time. Can you estimate how long it would have taken for the entire measurement of 160000 electrons?

[Note: The length of the electron wavepacket was about 1 micron. You will learn in Chapter 2 what this means, and why this is important in ruling out the presence of two electrons at the same time being in the double-slit apparatus].

Estimate time interval between the launch of two successive electrons, knowing an estimate of their speeds.

PLEASE DO PART C ONLY

Answer 1

The electrons are charged particles and have a magnetic field associated with it. If two electrons are inside the slit at the same time, they wil cause interaction with one another. Also, the experiment demonstrated here is used to demonstrate the probabilistic nature of quantum particles.

The potential used to accelerate the electrons is

V = 50 kV

So, the energy of the electrons is

(kilo electron volt)

Using non relativistic equations,

The velocity of electron is given by

Since the distance between the two electrons is 150 km = 150000 m,

The time between two electrons is

t = distance/speed = 150000/(1.325*10^8) = 0.0011321 s

The entire measurement consists of 160000 electrons.

The time taken for this many electrons to reach the screen is

T = 160000*0.0011321 = 181.132 s