This Cheat Sheet is intended to supplement Quantum Physics For Dummies, 3rd edition, by Andrew Zimmerman Jones. It begins by reviewing some useful operators used in quantum mechanics calculations. Then it covers a useful method for solving the Schrödinger equation for the quantum wave function, and then how you can use that wave function to calculate probabilities in quantum physics.

At some point, your quantum physics instructor may want you to add time dependence and get a physical equation for a three-dimensional free particle problem. You can add time dependence to the solution forif you remember that, for a free particle,That equation gives you this form forBecausethe equation turns intoIn fact, now that the right side of the equation is in terms of the radius vector r, you can make the left side match:That’s the solution to the Schrödinger equation, but it’s unphysical.

At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle,You can write its general solution aswhere Ay and Az are constants.Becauseyou get this forwhere A= Ax Ay Az.The part i

Each quantum state of the hydrogen atom is specified with three quantum numbers: n (the principal quantum number), l (the angular momentum quantum number of the electron), and m (the z component of the electron’s angular momentum,How many of these states have the same energy? In other words, what’s the energy degeneracy of the hydrogen atom in terms of the quantum numbers n, l, and m?

The Schrödinger equation is one of the most basic formulas of quantum physics. With the Schrödinger equation, you can solve for the wave functions of particles, and that allows you to say everything you can about the particle — where it is, what its momentum is, and so on. In the following version of the Sch

When calculating the spin of an electron in a hydrogen atom, you need to allow for the spin of the electron, which provides additional quantum states. Given the following equation, where the wave function of the hydrogen atom is a product of radial and angular parts, you can add a spin part, corresponding to the spin of the electron, where s is the spin of the electron and ms is the z component of the spin: The spin part of the equation can take the following values: Hence, now becomes And this wave function can take two different forms, depending on ms, like this: In fact, you can use the spin notation, where For example, for you can write the wave function as And for you can write the wave function as What does this do to the energy degeneracy?

Don’t think quantum physics is devoid of anything but dry science. The fact is that it’s full of relationships, they’re just commutation relationships — which are pretty dry science after all. In any case, among the angular momentum operators Lx, Ly, and Lz, are these commutation relations: All the orbital angular momentum operators, such as Lx, Ly, and Lz, have analogous spin operators: Sx, Sy, and Sz.

At some point, your quantum physics instructor may ask you to find the eigenfunctions of Lz in spherical coordinates. In spherical coordinates, the Lz operator looks like this: which is the following: And because this equation can be written in this version: Cancelling out terms from the two sides of this equation gives you this differential equation: This looks easy to solve, and the solution is just where C is a constant of integration.