Physical Explanation of Quantum Mechanics Notation? CLARIFICATION: I just don't understand what the notations below mean and how to use them.
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I just started taking QM, and the new notation is quite confusing.  While the math makes a nominal amount of sense during class lectures, I would find it much more satisfying to actually understand what the math means (and this would also assist in learning what operations and manipulations I need to do in a given situation).  Currently, I'm lost when I try doing any homework.
So, what are the physical interpretations of the following (if they exist):


*

*$\left| \psi \right\rangle$

*$\left\langle \psi \right|$

*$\left| \psi \right\rangle\left\langle \psi \right|$

*$\left\langle \psi | \psi \right\rangle$

 A: We can talk about what the notation represents. Terms I introduce will be italicised. For 1, the ket $\left|\psi\right\rangle$ represents the state of a physical system. Quantum mechanics claims these are elements of a vector space. So far, it's all physical. However, everything afterwards will abstract from that.
For 2 and 4, the bra $\left\langle\psi\right|$ represents the linear map $L$ from such states to complex numbers satisfying $\left|L\left|\phi\right\rangle\right|\leq L\left|\psi\right\rangle\in\left[0,\,\infty\right]$. For physical states we in fact impose $\left\langle\psi|\psi\right\rangle =1$. Computing an inner product is analogous to defining coordinates of points in space. (In fact, doing that can be written in this bra-ket formalism, but the usual notation on $\mathbb{R}^n$ is much easier.) If you know the axioms that define a vector space, you'll notice the set of bras is a vector space (as is the set of kets). It's called the dual space of the vector space of kets. In the bra-ket formalism, the two vectors in a "dot product" are thought of as different types of mathematical object; in a less pedantic formalism, the bras are just functions that take the dot product with a fixed vector.
The complex numbers obtained from bras are called inner products. For a generalisation of 3, the outer product $\left|\psi\right\rangle \left\langle\Psi\right|$ denotes the linear map $M$ satisfying $\left\langle\phi\right|M\left|\Phi\right\rangle=\left\langle\phi|\psi\right\rangle \left\langle\Psi|\Phi\right\rangle$. Note this is two inner products multiplied together, so $M$ is a map from a bra and a ket to a complex number. (In layman's terms, it's a matrix.) It's linear in the ket and antilinear in the bra, so it's called a sesquilinear map.
A: I think the easiest way to think about these objects is as follows:


*

*$|\psi\rangle$ is your physical state

*Your physical state comes with a machine (its dual) $\langle \psi |$, which when applied to any other physical state $|\phi \rangle$, spits out the overlap $\langle \psi | \phi \rangle$ between your state and $|\phi\rangle$

*It also comes with a projection $|\psi\rangle\langle \psi | $, which projects other physical states onto your state. E.g. acting on $|\phi\rangle$ by the projector gives $|\phi\rangle \to \langle \psi |\phi\rangle |\psi\rangle $

*$\langle\psi|\psi\rangle = 1$, because we like states to be normalised!


Hope that aids intuition. In more advanced treatments and for quantum computing, the projection, (3), is often the best way to get a handle on what's going on. 
A: A state $|\psi\rangle$ in quantum mechanics is a piece of information about a system that can be copied. $|\psi\rangle\langle\psi|$  and $\langle\psi|$ are both best understood as alternate representations of that same state. And $\langle\psi|\psi\rangle$ is a complex number such that $|\langle\psi|\psi\rangle|^2$ is the probability of the state $|\psi\rangle$ when the system is in the state $|\psi\rangle$, and $|\psi\rangle$ is often normalised so that $|\langle\psi|\psi\rangle|^2 =1$. 
Part of the problem with your question is that it puts the cart before the horse. You need to have a rough idea of what the appropriate explanation is before the maths will make much sense. If you want to understand quantum mechanics in greater depth, then read papers like http://arxiv.org/abs/1212.3245 and http://xxx.lanl.gov/abs/quant-ph/0104033 and lectures like http://www.quiprocone.org/Protected/DD_lectures.htm.
