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Why from Φ eq in page 136, he did not derive the solution in this form $Ae^{imφ}+Be^{-imφ}$https://drive.google.com/open?id=1thokDDIVDytck1-2HI3qyiG07mAMO1nc
hoalacanhdk said:But why he didn't use the combination of two solutions simultaneously? I think exp(-1)+exp(1) is a different solution from either exp(-1) or exp(1)
vanhees71 said:Well, again it seems to me the best advice is to use another QM textbook ;-)). If you look for eigenvalues of ##\hat{L}_z=-\mathrm{i} \hbar \partial_{\varphi}## you get unique solutions ##u_m(\varphi)=\exp(\mathrm{i} m \varphi)/\sqrt{2 \pi}## with the eigenvalues being ##m \hbar##.
A short answer, why ##m## should be integer is that you want uniqueness of the wave functions under rotations by ##2 \pi## aroud the ##z## axis. This is an incomplete argument however, because in quantum mechanics the absolute phase doesn't count. A more complete answer is that you get a complete orthonormal set of functions by this set when ##m \in \mathbb{Z}##.
If you look for the eigenfunctions of ##\hat{L}_z^2## from this it's clear that for each eigenvalue (except 0) the eigenvalue is degenerate, i.e., for each possible eigenvalue you have two linearly independent solutions, which you can choose of course as the eigenstates of ##\hat{L}_z##, i.e., ##u_{\pm m}(\varphi)## for each ##m## (with the eigenvalue ##\hbar^2 m^2## for ##\hat{L}_z^2##). You can also choose ##\cos(m \varphi)## or ##\sin(m \varphi)##. Of course, all these sets form again a complete set of orthogonal functions on ##\mathrm{L}^2([0,2 \pi])##, but why you would bother the students with a case of degenerate eigenvalues where you can much simpler treat the problem by first looking for the eigenvalues and eigenstates of ##\hat{L}_z##, I can't tell :-((.
The Φ equation in Griffiths' Introduction to QM is a mathematical expression that represents the wave function of a quantum mechanical system. It is a key equation in quantum mechanics and is used to describe the probability amplitude of finding a particle at a particular location and time.
The Φ equation is important in quantum mechanics because it is used to calculate the probability of finding a particle in a specific location and time. This equation helps us understand the behavior of particles on a microscopic level and is essential in predicting the outcomes of experiments in quantum physics.
The Φ equation is derived from the Schrödinger equation, which is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. The Φ equation is a solution to the Schrödinger equation and is obtained by applying certain mathematical techniques and boundary conditions.
The Φ equation has two key components: the wave function (Φ) and the Hamiltonian operator (H). The wave function represents the state of a quantum system, and the Hamiltonian operator describes the energy of the system. The equation also includes the time variable (t), which is used to calculate the time evolution of the system.
The Φ equation has many practical applications, including predicting the behavior of particles in quantum systems, calculating the energy levels of atoms and molecules, and understanding the properties of materials on a microscopic level. It is also used in developing technologies such as quantum computing and quantum cryptography.