Quantum mechanics textbook for 2nd year student

[jqmhelios]

I desperately need a good resource for quantum mechanics. This semester, our lectures have been absolutely terrible. Though I managed to avoid this by studying hard, the lectures and notes were so critically inadequate that most students (not including me) used ChatGPT to complete any online quiz.

I have Griffiths and Rae, but with the latter, I find it very verbose and lacking in concise explanations of what I need (even though the lecturer copied it word for word), and the former is mostly too advanced for my level.

Are there any other good quantum mechanics textbooks? I feel without these, my grade for this module will be exceptionally low

 

[Frabjous]

Here are some books TO LOOK AT to see if they speak to you:

Townsend https://www.amazon.com/dp/1891389785/?tag=pfamazon01-20
Schwabl https://www.amazon.com/dp/3540719326/?tag=pfamazon01-20
Shankar https://www.amazon.com/dp/0306447908/?tag=pfamazon01-20
Zwiebach https://www.amazon.com/dp/B0997R9CJ5/?tag=pfamazon01-20

Some out of print books
Liboff (I used it, but it has a mixed reputation)
McIntyre

I am unfamiliar with it, but it is a Dover book (i.e., cheap)
Park https://www.amazon.com/dp/0486441377/?tag=pfamazon01-20

As an undergrauate, I supplemented my QM texts with a more advanced text
Messiah https://www.amazon.com/dp/048678455X/?tag=pfamazon01-20

Browse the site. There are plenty of threads on this topic.

 

[PeroK]

I have Griffiths and Rae, but with the latter, I find it very verbose and lacking in concise explanations of what I need (even though the lecturer copied it word for word), and the former is mostly too advanced for my level.

Are there any other good quantum mechanics textbooks? I feel without these, my grade for this module will be exceptionally low

At the risk of stating the obvious, you need a textbook that aligns with the questions you'll face in any exams or course work. Griffiths, for example, does not use Dirac notation and focuses first on Wave Mechanics. You ought to go with a textbook recommendation for the course you are taking. Are you studing wave mechanics? - infinite square well etc.

I'm tempted to say that Griffiths is about as basic as it gets - although his style doesn't suit everyone.

Looking at Townsend, I see it starts with quantum spin. That's no good unless your course follows the same approach. And, it's explicitly for an "upper division" course.

Shankar must be more advanced than Griffiths. The best reviews for Shankar are from graduate students.

It's tricky, given that you must be well into your course by now.

 

[vanhees71]

If you want a very good textbook in the old-fashioned wave-mechanics-first approach, I'd recommend Messiah or Landau&Lifshitz vol. 3. My favorite still is J. J. Sakurai, Modern Quantum Mechanics, Revised Edition (also the newer edition extended by Napolitano is fine, but I dislike the addition being "Relativistic Quantum Mechanics", which is a contradictio in adjecto, since relativistic QT is in fact relativistic QFT). That's however also using the Dirac Bra-Ket formalism right away. It's a bit more abstract in the beginning, but the concepts get much more clear than with the old-fashioned "wave mechanics" approach.

 

[Vanadium 50]

I like Landau, but I am not sure I would want that as my first book. For example, Landau states that the natural coordinate system for tyhe hydrogen atom is...<drumroll>...parabolic.

Landau is 100% right. But as a beginner, that is far from obvious.

 

[vanhees71]

Parabolic coordinates are indeed great for the H atom, particularly to treat the scattering problem too as well as the Stark effect. The true thing is of course to use the Runge-Lenz vector and no "coordinates" at all to begin with ;-)).

 

[Vanadium 50]

Parab;oic coordinates are great. The fact thgat SO(4) breaks to SU(2)xSU(2) is explicit, which explains the k-l degeneracy: the radial eignenergies must look like the angular eigenenergies.

However, I wouldn't start from there.

 

[vanhees71]

Well, I'm also a bit unsure, how to treat the hydrogen atom for first-learners. At the moment I do first the angular momentum eigenvalue problem (i.e., finding the common eigenvectors of $\hat{\vec{J}}^2$ and $\hat{J}_3$ from the commutator relations alone, leading to $j \in \{0,1/2,1,\ldots \}$ and $m_J \in \{-J,-J+1,\ldots,J-1,J \}$ with eigenvalues $j(j+1) \hbar^2$ and $m_J \hbar$.

Then I argue that for the orbital angular momentum $\vec{L}=\hat{\vec{x}} \times \vec{p}$ there are only integer solutions, because of the meaning of the rotations $\exp(-\mathrm{i} \hat{\vec{L}} \cdot \vec{\phi})$ on the position eigenfunctions. Alternatively one can also argue with the 2D symmetric harmonic oscillator, where you can directly proove that $\hat{L}_3$ has only eigenvalues $m \hbar$ with $m \in \mathbb{Z}$.

Finally, I write down the time-independent Schrödinger equation and solve for the radial wave function in
the ansatz $\psi_{n\ell m}=R_{n \ell}(r) \text{Y}_{\ell m}(\vartheta,\varphi)$. This is, however, a quite lenghty calculation, starting with a discussion of the boundary conditions for $r \rightarrow 0$ and $r \rightarrow \infty$ and then the result that the energy eigenvalues are $E_n=-1 \text{Ry}/n^2$ with $n \in \mathbb{N}$. Then you have the somewhat astonishing additional degeneracy of the energy eigenvalues, i.e., that $E_n$ and $R_{n \ell}$ in fact only depend on $n$ and not on a combination of $n$ and $\ell$, which is specific for the Coulomb potential. Nevertheless it's a nice non-trivial example for the solution of a Sturm-Liouville boundary-value problem of a differential equation and the definition of the corresponding CONS of eigenfunctions.

This is well explained when using the Runge-Lenz vector which is known from classical mechanics as an application of Noether's theorem. The only trouble is that of course you need anglar-momentum addition, i.e., the Clebsch Gordan coefficients, because indeed the SO(4) symmetry (for the bound states) is solved by using $\mathrm{so}(4) \equiv \mathrm{su}(2) \oplus \mathrm{su}(2)$.

Another advantage of this analysis is that you also get the scattering states for $E=0$ (with the symmetry group ISO(3)$and$E>0## (with the symmetry group SO(1,3)).

 

[Vanadium 50]

All of this is good stuff, but I am not sure this is where I would start with a beginner.

FWIW, I prefer to start with k rather than n. The radial degree of freedome can do whatever it wants and the angular degrees can do whatever they want.

 

[vanhees71]

Interesting. So how would you present the H atom to beginners?

 

[Vanadium 50]

The first time? Separate and solve the Schroedinger Equation, Start with the angular piece and then introduce a centrifugal barrier, If the class would be more enlightened than confused, I might mention there are symmetries that can be exploited, but I am not going to use these tricks the first time through.

 

[SebasHerbert]

Typically, your teacher's recommended textbook should include about 80% of the exam material, and that should be enough. During our Quantum Mechanics course, we often switched between two textbooks: by Messiah та by J.J. Sakurai. Naturally, if you're feeling a little behind, you can always seek out additional information on resources like https://edubirdie.com/docs/, our teacher sometimes suggests looking beyond just textbooks, encouraging us to find lectures on topics that piqued our interest. Sometimes, these can be easier to track down.