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Q1:

Ok so i have come to a point where i know that $\Psi(r,t)$ which we denote only by $\Psi$ can be represented in a Hilbert space by a vector which we denote $\left|\Psi\right\rangle$. Does this mean $\left| \Psi(r,t) \right\rangle$?

Q2:

I know that equation $\Psi = \psi e^{- iWt/\hbar}$ represents a link between a stationary Schrödinger equation and a time dependant Schrödinger equation (we denote $\Psi(r)$ as $\psi$). I want to know now if we denote $\psi$ in a Dirac notation as $\left|\Psi(r)\right\rangle$ or any differently?

Q3:

How do we write a Schrödinger equation and a time independant Schrödinger equation using a Dirac notation?

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Q1/2. In Dirac notation, one does not usually write expressions like $|\psi(x,t)\rangle$ because the ket symbol denotes an element of a Hilbert space, not its corresponding representation in a particular basis. One does, however write expressions like $|\psi(t)\rangle$ to denote the state of the system at time $t$. If you wanted to write such a state in the position basis $\{|x\rangle\}$, then you would write $$ \psi(x,t) = \langle x|\psi(t)\rangle $$ Q3. In Dirac notation, the Shrodinger equation of time-evolution would be written as $$ i\hbar\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle $$ where $H$ is the Hamiltonian operator. The "time-dependent Shrodinger equation" is really just a (really bad my opinion) name for the eigenvalue equation for the Hamiltonian which, in Dirac notation, would be written as follows: $$ H|\psi\rangle = E|\psi\rangle $$

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  • $\begingroup$ 1st: You never used symbol $\Psi$. Why is so? Is this common praxis? 2nd: Does a $\left|\psi\right\rangle$ in a $\hat{H}\left|\psi \right\rangle = W \left|\psi \right\rangle$ means a time independant wave function? 3rd: I allso thought that if i act with a position operator $\hat{x}$ on a ket $\left|\psi(t)\right\rangle$ i can denote this as $\hat{x}\left| \psi(t)\right\rangle$ why did you write this as $\left\langle x | \psi(t)\right\rangle$? 3rd Does $\left\langle x | \psi(t)\right \rangle$ denote a wavefunction represented in position space/basis? Explain please in an edit. $\endgroup$
    – 71GA
    Commented May 20, 2013 at 6:03
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    $\begingroup$ @71GA 1st The making a notational distinction between $\Psi$ and $\psi$ is a matter of taste. Some authors use the former to indicate extra time-dependence, I prefer not to. 2nd Yes. 3rd $\langle x\psi(t)\rangle$ denotes the inner product of a position basis vector and another vector, not the position operator acting on the vector. Yes, the function $\psi$ defined by $\psi(x,t) = \langle x|\psi(t)\rangle$ is the position space representation of $|\psi(t)\rangle$. $\endgroup$
    – joshphysics
    Commented May 20, 2013 at 6:14
  • $\begingroup$ Can you please than explain what is the difference in acting on a ket $\left| \psi \right\rangle$ with an operator $\hat{x}$ like this $\hat{x}\left|\psi\right\rangle$ or calculating the inner product $\left \langle x | \psi(t) \right \rangle$ - if possible, could you provide a physicall interpretation (after 3rd you have misstyped an inner product i think. There is a $|$ missing i think). $\endgroup$
    – 71GA
    Commented May 20, 2013 at 6:59
  • $\begingroup$ @71GA Acting on a vector with a linear operator gives another vector. Taking the inner product of two vectors gives a number. Unfortunately I can't give you a comprehensive treatment of linear algebra and its physical interpretations in the context of quantum mechanics here; I think you would really benefit from reading a systematic treatment of bra-ket notation; all of these concepts are covered in a lot of books and notes etc. Yes I mistyped at the end. $\endgroup$
    – joshphysics
    Commented May 20, 2013 at 7:37
  • $\begingroup$ Have you read Zetilli's book? Is the chapter "Postulates of QM" the thing i need to read? It sure looks nice... $\endgroup$
    – 71GA
    Commented May 20, 2013 at 8:48

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