In this video, we will define particle current. In the potential step and barrier problem that we have dealt with earlier, we calculated the transmission and reflection coefficients from the ratios of the probability amplitudes. Now, this is perfectly fine if we are dealing with actual waves, as in optics and acoustics. But in quantum mechanics, we are actually dealing with particles that behave like waves. Is it possible to define actual particle current density, no flux of number of particles per time? The particle current is related to the rate of change in particle density via the continuity equation, which is this. In this equation, what it's saying is that J here is the particle flux or current density, and the divergence of J basically tells us the net rate of particles flowing in and flowing in out. That, as long as we're not creating or annihilating particles, should be directly related to the rate of change of the density of particles in that volume. That's what this continuity equation is telling us. In quantum mechanics, the particle density is given by the absolute value square of the wave function, because that's basically the probability of finding a particle. Time-dependent Schrodinger equation gives you the time evolution of that wave function. Once again, Hamiltonian is given by the standard equation. The specifics of the problem is given usually by the shape, the functional form of this potential energy. Now take a complex conjugate of the time-dependent Schrodinger equation as shown here, and then you multiply the complex conjugate of the wave function to Equation 1, this guy, minus you multiply the wave function Psi to this equation and take a subtraction here. Then the left-hand side, this, this combined gives you simply the ih-bar times the time derivative of the quantity Psi start Psi, which is the absolute value square. The right-hand side, as long as the potential is real, potential goes away and you end up with this, which from the vector calculus identity, you can write it like this, is h-bar squared over 2m divergence of this quantity in the parenthesis. Now we're going to define the particle current density J to be proportional to this quantity in the parenthesis. Specifically, you multiply ih-bar divided by 2m to it. The reason that we do that is if we do that, then the equation that we derived from time-dependent Schrodinger equation simply turn into the continuity equation, because this is simply the density. The other term, this term here gives you the divergence of the current density. Now, let's see if it makes sense by considering a couple of examples. First potential step problem, the solutions were this, this is the left side of the potential step, if you recall, this is the incident wave, this is the reflective wave. This here is the wave function to the right of the potential step, so this represents the transmitted wave. Now let's calculate the particle current density for the incident wave, which is this. Just use the definition of particle current density, then you get this A square amplitude. The amplitude squared times h-bar k naught over m. What is this? h-bar times k naught represents the momentum divided by m is v, so it's a velocity and A, absolute error square represents what? The density. This here represents the particle flux as the particle flux J, the particle current density should. Similarly, reflected particle flux or current density is given by this. Once again, h-bar times k naught divided by m is the velocity, and there is a negative sign because it's going backwards in the negative z-direction. Then this here is A squared, is related to the particle density of the incident, and this gives you the ratio or the fraction of the incident particle that are being reflected. In the transmission case, this time we have h-bar times k divided by m, this is the velocity of the transmitted particle, and this again represents the ratio of the transmitted particle to the incident particle density. Now let's consider another example of potential well problem. Infinite potential well solution is given by this. Now if you calculate the current density, well, this wave function is real, and therefore, Psi star and Psi are the same, and so these two terms inside the parentheses are the same, and therefore, it gives you a zero flux. Now, you have a confined system here. Electrons are confined within a potential well, they're not going anywhere, so you should have reasonably expected zero particle flux, and that's what you get. One should expect zero particle flux in all confined system. For example, harmonic oscillator problem, finite potential well problem, if you're dealing with a confined system, you should expect zero particle flux or particle current density, and that means that your wave function is always real. Complex conjugate and the function itself is the same, so we should expect purely real wave functions in a confined system only when you have an unbound system like potential step or potential barrier problem, only then you should expect complex wave function.
