Undergrad Understanding the Wave Function of Electrons: From Theory to Experiment

[rahaverhma]

1)Is the wave function of the electron perpendicular to the motion of electron in straight line in the similar fashion as that of the photons.?
2) And what is the origin of this wave function?
3) can someone give me the details about the electron in double slit experiment (and mainly a theory abt the every single electron released one at a time with two slits open, and observing it after the hitting on screen)

 

[vanhees71]

I've no clue what you mean with 1). The wave function in non-relativistic quantum theory is a probability amplitude for finding the electron at a given place, i.e., the probability distribution for finding the electron at place $\vec{x}$ when measured at time $t$ is given by
$$P(t,\vec{x})=|\psi(t,\vec{x})|^2.$$
It evolves according to the Schrödinger equation,
$$\mathrm{i} \hbar \partial_t \psi=\hat{H} \psi,$$
where $\hat{H}$ is the Hamiltonian of the electron. For a free electron, i.e., no external potentials or fields present, you have
$$\hat{H}=\frac{\hat{\vec{p}}^2}{2m},$$
and you can solve the Schrödinger equation in that case analytically in momentum representation and then build wave packets in position representation via Fourier transformation. What comes out is that the expectation value of the position is a straight line with constant velocity (uniform motion) as in classical physics, but the wave function tells you that the standard deviation grows with time, which is due to the position-momentum uncertainty relation, which is implied by QT.

Ad 2) I don't know, what you mean by "origin of the wave function". Within the position representation it's the basic building block of QT. You cannot explain it from anything else more simple.

Ad 3) The double-slit experiment is solved by just solving the corresponding boundary-value problem for the Schrödinger equation, leading to interference effects pretty analogous to the (somewhat more complicated) case of electromagnetic waves. However, again, the interpretation of the result for the Schrödinger equation is completely different from that of solving the Maxwell equations: The Schrödinger wave function gives probabilities for finding an electron at a given position at the screen of detection. Each electron just leaves a single spot on the screen. Only with many (equally prepared) electrons running through the slits and making a dot on the screen leads to the interference pattern.

 

[jfizzix]

For 1, the wavefunction of the electron is a scalar, rather than a vector, so it has magnitude, but no direction. Even though it has real and imaginary parts, it doesn't have cartesian (x,y,z) components. It doesn't point in any direction at all.

For 2, The wavefunction came from the minds of brilliant theoretical physicists as a compact way for representing all the measurable information about a quantum system. Its objective reality is still the subject of debate, and there's no broad consensus, though it certainly works very well.

For 3, the double slit experiment shows that electrons have both wave-like and particle-like properties. Even though each electron hits a single point on the screen, the interference pattern that emerges can only be explained (so far) by a wave-like description. Also, any measurement device that can obtain information about which slit the electron went through has to disturb the electron enough that the wave-like interference pattern is destroyed.