Problem 1.1 Consider a particle and two normalized energy eigenfunctions ψ1(x) and ψ2(x) corresponding to the eigenvalues E1 = E2. Assume that the eigenfunctions vanish outside the two non-overlapping regions 1 and 2 respectively. (a) Show that, if the particle is initially in region 1 then it will stay there forever. (b) If, initially, the particle is in the state with wave function ψ(x, 0) = √ 1 2 [ψ1(x) + ψ2(x)] show that the probability density |ψ(x, t)| 2 is independent of time. (c) Now assume that the two regions 1 and 2 overlap partially. Starting with the initial wave function of case (b), show that the probability density is a periodic function of time. (d) Starting with the same initial wave function and assuming that the two eigenfunctions are real and isotropic, take the two partially overlapping regions 1 and 2 to be two concentric spheres of radii R1 > R2. Compute the probability current that flows through 1.
acid battery, measuring its voltage is the easiest way to tell how charged it is and from that data, the state of charge can be worked out. With a lithium-ion battery, it is slightly difficult to work out making it very hard to tell the state of charge of a cell as the voltage relative to the state of charge is very flat around 60% of the charge/discharge curve with sharp inflections at both ends [8]. This makes it relatively easy to cause permanent damage by both over and undercharging the cell and damaged cells can overheat and get into a thermal runaway condition [8]. This is where the BMS plays a crucial role in managing risk and ensuring safety.
The BMS accurately reports the state of charge of the battery pack and monitors the cells detecting any problems to ensure that the pack can be safely shut down when needed. The BMS consists of several separate circuit boards typically such as module control units or MCUs which will monitor the voltage of a small group of cells or individual cells very accurately in the battery. The MCU will also include some temperature measurements to monitor inside of the battery pack and possibly the individual cells linking up to a master BMS module which will combine data from all the slave MCUs measuring overall current flow in and out of the battery pack. With this information, it can work out what the battery state of charge is and reported to the driver via a display or warning light(s) [8].
Ability to charge each cell up to its maximum safe voltage and discharge it down to its minimum safe voltage limits the battery pack capacity. Cell tolerance in the manufacturing processes gives them slight variations in how quickly they will charge and discharge. The BMS also balances the cells making sure that the battery pack is not limited to the capacity of the cell that gets emptied and filled quicker by using charge pumps or b