I'm looking for a text to complement Frederic Schuller's lectures on QM. His approach is very mathematical — in fact it looks like the first 12 of 21 lectures are just about the mathematical foundations of QM. He introduces Hilbert and Banach spaces from scratch (from basic linear algebra and analysis really), derives the functional analysis version of the spectral theorem, and gives what I assume are more rigorous definitions. For instance in all of the undergrad books I've looked at, quantum states — if they're given any definition at all — are said to be elements of the Hilbert space. But Schuller claims that that is not correct. States are in fact positive trace-class linear maps on the Hilbert space. Does anyone know a good undergrad level QM book that I can follow along with so I have some exercises and extra material to work through as I go through the lectures? Thanks.
[Math] Complementary text for mathematical Quantum Mechanics lectures
book-recommendationfunctional-analysismathematical physicsquantum mechanicsreference-request
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As user953376 pointed out in their comment one-dimensional subspaces are determined by a unit vector. To fill in some detail: as an application of the Riesz representation theorem for every bounded linear operator $T$ on $\mathscr H$ the image of which is a one-dimensional, there exist $\psi,\phi\in\mathscr H$ such that $T=|\phi\rangle\langle\psi|$ (that is, $T(z)=\langle\psi,z\rangle\phi$ for all $z\in\mathscr H$). Now if $T$ is a self-adjoint projection then $T=|\psi\rangle\langle\psi|$ and $\langle\psi,\psi\rangle=1$. Thus every pure state in the above sense is determined by a unit vector, giving rise to the mapping \begin{align*} \iota:\{\psi\in\mathscr H:\langle\psi,\psi\rangle=1\}&\to\{T\in\mathcal B(\mathscr H):T\text{ orthogonal projection},\operatorname{dim}(\operatorname{ran}(T))=1\}\\ \psi&\mapsto|\psi\rangle\langle\psi|\,. \end{align*} While $\Psi$ is surjective, it fails to be injective (again for the reason user953376 described in their comment): $\psi$ and $e^{i\alpha}\psi$ are mapped to the same operator under $\iota$. This is solved by replacing the domain of $\iota$ by equivalence classes $[\psi]$ under the equivalence relation $\psi\sim\psi':\Leftrightarrow \psi=\psi' e^{i\alpha}$ for some $\alpha\in\mathbb R$---then $\iota$ is the isomorphism you were looking for.
As for your second question: there are two main reasons to consider states to be positive semi-definite trace-class operators of unit trace---or, equivalently, positive normal functionals $f$ on a $W^*$-algebra such that $f({\bf1})=1$---as opposed to mere elements in a Hilbert space:
- As seen before, defining states via vectors would mean that global phases make a difference (i.e. $\psi\neq e^{i\alpha}\psi$ for any $\alpha\in(0,2\pi)$). But such phases can never be detected experimentally because they vanish in the expectation value $\langle\psi|A|\psi\rangle$ and thus in the measurement.
- Secondly and more importantly, looking at trace class operators allows for a description of mixed states. Such states are convex combinations of pure states and usually arise whenever relaxation or other environmental effects enter the picture. For example the maximally mixed state for a qubit $\operatorname{diag}(1/2,1/2)=\frac12|0\rangle\langle 0|+\frac12|1\rangle\langle 1|$ describes that there is a 50% chance of the state becoming the first eigenstate ($|0\rangle\langle 0|$) after measuring it and, similarly, a 50% chance for the system to be in the second eigenstate ($|1\rangle\langle 1|$) after measurement. Now a description of states via elements of $\mathscr H$ falls short of such features for the eigenvalue (probability) of $|\psi\rangle\langle\psi|$ is always equal to one, so there is "nothing else" in $\mathscr H$ which could describe mixed states and obey a quantum-mechanical probabilistic interpretation.
To expand on this second point: every state $M$ (positive semi-definite trace class operator of unit trace) on an arbitrary Hilbert space allows for a spectral decomposition (because trace class) $M=\sum_{j=1}^\infty p_j|\psi_j\rangle\langle\psi_j|$ with "probabilities" $p_j\in[0,1]$ (positivity), $\sum_{j=1}^\infty p_i=1$ (trace condition) and an orthonormal system $\{\psi_j:j\in\mathbb N\}$ in $\mathscr H$. Therefore the pure states are fundamental to the concept of states as those are the "building blocks", but in order to get the full picture one has to allow for mixtures of such states, which are mathematically described by such convex combinations.
It is not strictly necessary to learn quantum mechanics (QM) as a prerequisite to quantum computing (QC), since QC questions can be framed purely in terms of linear algebra; typically about unitary and self-adjoint operators over finite dimensional Hilbert spaces.
In their full generality, the postulates of QM describe systems whose state space could be an arbitrary Hilbert space. However, in practice one tends to consider either finite dimensional Hilbert spaces (essentially $\Bbb C^n$) or the Hilbert space $L^2(\Bbb R^n)$ of square-integrable complex-valued functions over $\Bbb R^n$. In this second case, the state-vector $\psi = \psi(\mathbf r,t)$ (describing the state of the system at a given time $t$) is a square-integrable "wave function" over $\Bbb R^n$, and the relevant inner product is given by $$ \langle \phi | \psi \rangle = \int_{\Bbb R^n} \phi^*(\mathbf r,t)\psi(\mathbf r,t)\,d\mathbf r. $$ If one focuses exclusively on this second case, then it is fine to say that the state-vector is always (within the context of interest) a wave-function of this kind.
Best Answer
I'll just make my comments into an answer.
R&S Volume one introduces Hilbert spaces, Banach spaces, spectral theorem etc. and leads from bounded to unbounded operators and the fourier transform.
R&S Volume 2 is very physics orientated, with topics on fourier transforms, hamiltonians in non-rel QM, and talks about self adjoint operators, and a bit about time dependent Hamiltonians.
As a note on quantum states, there's various definitions. They can be
It depends on what you want to do with them.
The first I think is the most common, as when teaching quantum mechanics, the wave functions usually belong to an $L^2$ space, and are found by solving the Schrodinger PDE.
The second last one is useful for statistical mixtures and open quantum systems, you can have pure and mixed states. The pure states can be identified with the last item on the list.
As a sidenote this was asked by a different user on physics stack, same question on schullers course, and it was closed, even though it's physics, and a pretty reasonable request. It might be useful to check there for the one answer that was able to be posted before it was closed.
https://physics.stackexchange.com/questions/259583/good-texts-on-quantum-mechanics-to-accompany-this-online-course#comment579079_259583