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9
votes
3answers
1k views

What is a basis for the Hilbert space of a 1-D scattering state?

Suppose I have a massive particle in non-relativistic quantum mechanics. Its wavefunction can be written in the position basis as $$\vert \Psi \rangle = \Psi_x(x,t)$$ or in the momentum basis as $$...
8
votes
1answer
822 views

Normalizing Propagators (Path Integrals)

In the context of quantum mechanics via path integrals the normalization of the propagator as $$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$ is incorrect. But why? It gives the correct pre-...
8
votes
2answers
447 views

How to guarantee square integrable solutions to time-independent Schrödinger's equation?

Given the time-independent Schrödinger’s equation in one dimension $$H\psi = E\psi$$ what restrictions can we place on V(x) (inside the hamiltonian) and E to guarantee that the solutions won't have ...
6
votes
2answers
171 views

Physical position eigenfunction normalisation

We know that the Dirac function $$\delta(a)=\lim_{a \rightarrow 0} \delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$ \delta_{a}(x) := \frac{1}{\sqrt{2\pi a^2}}e^{-x^2/2a^2}$$ ...
5
votes
1answer
232 views

S-Matrix and normalization of states

I'm trying to understand what is the S-matrix in QFT. People say that it has to be a unitary matrix, but that I guess will change with a different normalization of the incoming and outgoing states. My ...
4
votes
1answer
77 views

Hilbert space, does $|r\rangle$ satisfy $\langle r |r\rangle = 1$?

Let's say we start with no particles: $\mid0\rangle$. We have $\vec{p}\vert0\rangle = 0$, $H\vert0\rangle = 0$, where we are ignoring $\infty$ vacuum energy. Also, $a(\vec{k})\vert0\rangle = 0$ for ...
4
votes
1answer
220 views

Why must these Spinors be normalized?

I have just begun studying spin and there are two spinors mentioned: The main spinor $\chi $ and the spin-up spin down spinors (eigenspinors) $\chi_+ ,\chi_- $. I learned that the main spinor is a ...
3
votes
3answers
973 views

Who is doing the normalization of wave function in the time evolution of wave function?

In the Schrodinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations ...
3
votes
3answers
448 views

Normalization of wave function meaning…?

I just have one question. I'm doing a problem where I'm told to normalize a wave function, which is split up into two regions, namely where $r \leq r_0$ and $r > r_0$. My question is, why am I ...
3
votes
1answer
213 views

Expected value inequality

Why is $\langle p^2\rangle >0$ where $p=-i\hbar{d\over dx}$, (noting the strict inequality) for all normalized wavefunctions? I would have argued that because we can't have $\psi=$constant, but ...
3
votes
1answer
152 views

What does the notation $\Psi_k/(\Psi_k,\Psi_k)^{1/2} $ mean?

I am currently reading the paper "Gravitation and quantum mechanics for macroscopic objects" by F. Karolyhazy (1966). In his paper, he uses certain notation that I haven't come across before (he also ...
3
votes
2answers
81 views

Understanding operator bra-ket notation

Hi I have a question that might be a bit trivial. I have just completed learning a section on the bra-ket notation. There is a statement that the following is prohibited $$\hat{A}\langle\psi|, ~|\psi\...
3
votes
1answer
83 views

Normalising a wave function in parts?

If we have the wave function $\psi_{100}(r,\theta,\phi)=R_{10}(r)Y_{00}(\theta,\phi)$ when we are normalising it we do the following: $$1=\int| \psi_{100}(r,\theta,\phi)|^2sin(\theta) r^2drd\theta d\...
3
votes
1answer
81 views

Does it really make sense to talk about field lines?

Field lines should only provide a visual representation of a field. There is a rule for their construction: take an object subject to a field, move it by d$\mathbf{r}$ and draw the direction of the ...
3
votes
1answer
100 views

Free space propagator: reconciling two results

In quantum mechanics, the free space propagator $G(q_f=0,q_i=0;\tau)$ can be easily calculated to be $$\sqrt{\frac{m}{2\pi i \hbar \tau}}$$ by inserting an identity operator. However if we use ...
3
votes
0answers
38 views

Why is the inner product of position eigenstates not normalised? [duplicate]

I have read that $$<{\bf r}|{\bf r}'> = δ({\bf r}-{\bf r}').$$ I don't understand how this is correct, I want to say it is equal to 1 or 0, rather than an unnormalised delta function. Clearly ...
2
votes
3answers
404 views

Normalization problem with hydrogen wavefunction

Suppose you have a mix of states made up of the Hydrogen $\lvert nlm \rangle$ states where one of the coefficients is unknown. For example: $$ \lvert \psi\rangle=A\lvert 100\rangle + \sqrt{\frac{2}{3}}...
2
votes
3answers
678 views

Normalization of basis vectors with a continuous index?

I have an infinite basis which associates with each point, $x$, on the $x$-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ ...
2
votes
1answer
265 views

Normalizing wavefunction

If you are trying to normalize $\psi = A\sin kx$, and you find that $|A|^2 = \frac{2}{a}$, why do you take the positive square root and not the negative? What happens if you take the negative square ...
2
votes
3answers
2k views

Wave function not normalizable

Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\psi(x)$ $$\int_{-\infty}^{\infty}|\psi(x)|^2 dx<\infty \qquad \text{or}...
2
votes
3answers
300 views

Expectation Values and Derivation of Heisenberg Equation?

Consider a system of particles with wave function $\psi$(x) (x can be understood to stand for all degrees of freedom of the system; so, if we have a system of two particles then x should represent {$...
2
votes
2answers
112 views

Normalisation of free particle wavefunction

The wavefunction $\Psi(x,t)$ for a free particle is given by $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$ This wavefunction is non-normalisable. Does this mean that free particles do not exist in ...
2
votes
1answer
161 views

Normalization of potential barrier solution

I don't understand a point in the solution attached to this barrier potential problem. Below equation 4.209, they say Assume first that the wave function on the right side of the barrier in the ...
2
votes
1answer
1k views

Superposition of wavefunctions

Suppose you have 2 normalized wavefunctions $\psi_1=Ne^{iax}e^{if(x)}e^{i\omega t}$ and $\psi_2=Ne^{-iax}e^{if(x)}e^{i\omega t}$ defined on $x\in [-x_0,x_0]$ and vanishes for $|x|>x_0$. What then ...
2
votes
2answers
4k views

Why do wave functions need to be normalized? Why aren't the normalized to begin with? [duplicate]

Before I started studying quantum mechanics, I thought I knew what normalization was. Just pulling off Google, here's a definition that matches what I've understood normalization to mean: ...
2
votes
2answers
252 views

quantum mechanics operators - Hermitian or complex conjugate?

Let $f(x)$ be a normalised state in a 1-D system. Let $g(x) = iA f(x)$, where $A$ is a Hermitian operator. I want to find the inner product of $g(x)$ with itself. Is it $$\int \left(-i A^\dagger f^*(...
2
votes
2answers
90 views

Normalization of a wave function in quantum mechanics

A more simple question, so I am watching a quantum mechanics lecture on potentials of free particles and am doing the general solution of schrodinger's stationary equation for a free particle when I ...
2
votes
1answer
303 views

Green's Functions from Gell-Mann and Low Theorem

What I want to do: $\newcommand{\ket}[1]{\left|#1\right\rangle}$ $\newcommand{\bra}[1]{\left\langle#1\right|}$ $\newcommand{\braket}[1]{\left\langle#1\right\rangle}$ The Gell-Mann Low Theorem tells ...
2
votes
1answer
69 views

Normalizing a wave function in a mixed well

So I got this potential and want to solve for the even wavefunctions http://imgur.com/GKAy4nD Since it's symmetric around the origin I need only to look at the interval [0,b] and solve for the ...
1
vote
5answers
3k views

Normalizing the solution to free particle Schrödinger equation

I have the one dimensional free particle Schrödinger equation $$i\hbar \frac{\partial}{\partial t} \Psi (x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \Psi (x,t), \tag{1}$$ with ...
1
vote
2answers
402 views

Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
1
vote
1answer
196 views

Is there a normalized form of the Euler equation discretized with finite volumes?

I want to calculate a flux on my fpga using the Euler equations with the finite volume method. Unfortunately the values of the state variables differ a lot. For example the pressure has a value of ...
1
vote
1answer
917 views

How do we normalize a delta function position space wave function? [duplicate]

I have a position space wavefunction $$\psi(x) = \delta(x-a) + \delta(x+a).$$ Now the question states to compute the following: The Fourier transform of $\psi(x)$. (Which invariably is the momentum ...
1
vote
1answer
61 views

Normalize Triplet State of Hydrogen

For hydrogen, the total spin of the electron and proton is $s = 1$, while $m_s = -1,0,1$. If $m_s = 1$, one of the states can be written as $$\left| 1\;1 \right > = \left |\uparrow \uparrow\right &...
1
vote
1answer
244 views

Normalizing continuous eigenstates

As far as I understand, to normalize the eigenfunctions, corresponding to the continuous spectrum, we use Dirac delta function: $\langle \psi_\lambda \mid \psi_{\lambda'} \rangle = \delta(\lambda - \...
1
vote
1answer
68 views

Spin half for the value of $|1 0\rangle$?

Spin-1/2 The eigenspinor , $X=aX_++bX_-$ $$X_+=\left( \begin{array}{cc} 1\\ 0\end{array} \right) $$$$X_-=\left( \begin{array}{cc} 0\\ 1\end{array} \right)$$ They are define like this because they ...
1
vote
2answers
4k views

Proper notation for normalized scalar?

I have not been able to find a resource to tell me the standard notation for a normalized scalar value. Normalized vectors (i.e. unit vectors) are typically denoted by placing a hat over the variable,...
1
vote
1answer
4k views

Normalizable wave functions?

How can I test whether a wave function is normalizable? If you apply an operator to a wave function, sometimes the result will not be normalizable. But how can I find these wave functions that do not ...
1
vote
1answer
40 views

“Normalisation” in the unitary gauge

I will use the example of the Abelian Higgs model to explain my problem. Consider the Lagrangian: $ \mathcal{L} = - \frac{1}{4} F^{\mu \nu}F_{\mu \nu} + \left(D^\mu \phi\right)^\dagger \left( D_\mu \...
1
vote
1answer
176 views

Why don't we need to normalize wavefunction to find probability distribution?

Consider an unormalized wavefunction of a rotor at $t = 0$, a combination of $n=0$ and $n=2$ states: $$\psi(\phi) = 3 - 2 \cos (2\phi).$$ Find the probability distribution in angle. The ...
1
vote
1answer
552 views

Quantum Mechanics - Rectangular Potential Barrier - Normalisation

I have a quick question regarding the normalisation of the wave function of a particle incident on a potential barrier specifically regarding the normalisation of the wave functions. The problem is ...
1
vote
1answer
1k views

Orthogonality of the wavefunctions on an subinterval?

Lets say that functions (eigenfunctions) $\psi_0$ and $\psi_1$ are orthogonal on an interval $-d/2 < x < d/2$. Are they also orthogonal on any subinterval inside the interval $-d/2 < x < d/...
1
vote
0answers
48 views

How can we justify identifying the Dirac delta function with the eigenfunction of position? [duplicate]

I can think of at least two different ways to understand eigenfunctions of operators in quantum mechanics. But neither one seems to provide a good explanation for why we take the position-basis ...
1
vote
0answers
42 views

Help normalising and taking the inverse Fourier transform of this wavefunction [closed]

Normalising Consider the wavefunction $$\psi(x,0)=Ne^{-\frac{|x|}{\lambda}}.$$ In order to normalise this I take the integral, which due to the modulus on the $x$ I evaluate just from zero to ...
1
vote
1answer
521 views

Normalisation of Linear Harmonic Oscillator - Ladder Operator Method

I was watching the following video on the harmonic oscillator using ladder operators : http://youtu.be/gRdCV9p8sAU?t=30m9s Clicking on the video above will take you to the exact point where my ...
0
votes
1answer
138 views

Griffith's Proof that a wave function will stay normalized is incorrect?

In Griffith's book, Introduction to Quantum Mechanics, in the equation: $$ \frac{\partial}{\partial t} \left| \Psi\right|^2 = \frac{i\hbar}{2m} \left( \Psi^* \frac{\partial^2 \Psi}{\partial x^2} - \...
0
votes
1answer
3k views

Wavefunction normalization

How do we normalize a wavefunction that's a linear combination of sines and cosines (or of $Ae^{ikx}+Be^{-ikx}$ -- they're the same, right)? One you square it, wouldn't the integrand be oscillating ...
0
votes
2answers
634 views

Normalising a wavefunction where $\psi$ is equal to a sum of functions [closed]

The wavefunction $\psi(x)$ = $\phi_1(x)$ + $2\phi_2(x)$ + $3\phi_3(x)$ is to be normalised. The functions $\phi_1(x)$, $\phi_2(x)$, $\phi_3(x)$ are normalised eigenfunctions of a Hermitian operator $\...
0
votes
3answers
1k views

Normalisation factor $\psi_0$ for wave function $\psi = \psi_0 \sin(kx-\omega t)$

I know that if I integrate probabilitlity $|\psi|^2$ over a whole volume $V$ I am supposed to get 1. This equation describes this. $$\int \limits^{}_{V} \left|\psi \right|^2 \, \textrm{d} V = 1\\$$ ...
0
votes
2answers
72 views

$\newcommand{\b}[1]{\langle#1\rangle}$Is the expectation of an operator written as $\b{\psi|\hat A|\psi}$ or as $\b{\psi|\hat A|\psi}/\b{\psi|\psi}$?

I had presumed that the expectation of an operator is written as $\b{\hat A} = \b{\psi|\hat A|\psi}$, but some online reference insists on dividing the entire expression by $\b{\psi|\psi}$. Since $\b{\...