Too few examples to explain The principles of quantum mechanics by dirac.

[DiracRules]

Too few examples to explain "The principles of quantum mechanics" by dirac.

Hi!

I studied my first course of quantum physics without a technical formalism (I'm studying physics engineering).
I find some hindrances in paragraph 20.
It says (I'm translating from Italian):

In a representation in which the complete set of commuting observables \xi_1',\ldots,\xi_u' are diagonal any ket |P> will have a representative <\xi_1'\,\,\xi_u'|P> or <\xi'|P> for brevity. This representative is a definite function of the variables \xi', say \psi(\xi'). The function \psi then determines the ket |P> completely, so it may be used to label this ket, to replace the arbitrary label P. In symbols, if <\xi'|P>=\psi(\xi') we put |P>=|\psi(\xi)>

After a few lines, he says that we can formally write |P>=\psi(\xi), where \psi(\xi) is the wave function.

What I cannot understand is how to transpose these symbols in effective calculations.
For example, if |P> represent the superposition of the first two states of a particle (say an electron) in an infinite well, what is \psi(\xi)? How can I find it?

 

[Bill_K]

If we let |0> and |1> denote the ground state and first excited state of a system, and let |P> be some superposition |P> = α |0> + β |1>, then we can represent these quantities in terms of the position x by multiplying on the left by the bra <x|, getting <x|P> = α <x|0> + β <x|1>. Or as functions, calling <x|0> ≡ f₀(x) and <x|1> ≡ f₁(x) we have the wavefunction P(x) = α f₀(x) + β f₁(x).

The advantage to Dirac's bra-ket notation is that it is more general: you can represent everything in terms of functions of momentum by multiplying instead by <p|. Or you can handle more general cases in which the wavefunction is described by several quantities, such as spin plus position. Bra-ket notation also makes it easy to convert back and forth between different representations.

 

[ardie]

your kets, are a set of solutions to the wavefunction dscribed by the Hamiltonian, and can interpreted as physical states of the system. the corresponding eigenvalues are then the probabilities associated with finding the system in that eigenstate (provided we have an orthonormal basis set), but the notation allows you to transform your set of solutions to any observable (say momentum or spin state), using the orthogonolisation procedure.