The tag has no usage guidance.

learn more… | top users | synonyms

2
votes
2answers
67 views

Express state as eigenkets

This is very basic but I just suddenly got confused. Any state can be expressed as complete set of eigenkets with discrete eigenvalues: $$|P\rangle = \sum^n c_n |p_n\rangle$$ I understand the above. ...
0
votes
2answers
17 views

Normalization of time-independent Schroedinger equation in Spherical Coordinates

I have a short question about the time-independent Schroedinger equation in Spherical coordinates: $$\psi(r, \theta, \phi) = R(r)Y(\theta, \phi)$$ then the normalization condition becomes $$\int |\...
3
votes
0answers
44 views

Energy term in Wavefunction Normalization

I recently started learning quantum mechanics and when I solved the Schrödinger equation for the Hydrogen atom, in particular the Radial equation, I found that I had normalized it but a term in the ...
12
votes
2answers
986 views

How should Dirac notation be understood?

If vectors $|\vec{r}⟩$ and $|\vec{p}⟩$ are defined as $$ \hat{\vec{r}} |\vec{r}⟩ = \vec{r} |\vec{r}⟩ \\ \hat{\vec{p}} |\vec{p}⟩ = \vec{p} |\vec{p}⟩ $$ then one can see that products like $$ ⟨\vec{...
0
votes
1answer
25 views

Why must the separation constant be real in a time dependent wave function?

I'm not sure if I'm asking this right. I'm reading ''Introduction to Quantum Mechanics'' by Griffiths and in the chapter 2 exercises he asks to prove that the separation constant, $E$, must be real. ...
0
votes
0answers
35 views

Why Does the Dirac delta Function Fix the Normalization of the Basis Vectors in Infinite Dimensions? [duplicate]

On page 60 of Shankar's intro to QM at the very bottom he says that the Dirac delta function fixes the normalization of the basis vectors with an infinite amount of dimensions. I don't understand why ...
0
votes
1answer
32 views

Normalisation of angular wave function: particle in a circular box

For a particle in a circular box (with radius $R$) with zero potential inside the circle and infinitely high potential outside of the circle, the Schrödinger equation in polar coordinates is: $$-\...
1
vote
0answers
52 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 ...
0
votes
0answers
30 views

What is the significance of Dirac ortho-normality? [duplicate]

What is the significance of Dirac ortho-normality? We know for momentum eigenfunction $f(p,x)$ for eigenvalue $p$ , $$\langle f(p',x) | f(p,x)\rangle~=~ \delta(p - p') $$ I am not clear why it is ...
0
votes
1answer
35 views

Norm of quantum state in three dimensions

The Born interpretation states that for a particle with a wave function $\Psi(x)$, the total probability of finding that particle at some point in space is equal to $\int_{-\infty}^{\infty}\Psi(x)^*\...
3
votes
2answers
89 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\...
0
votes
1answer
37 views

Inner product of standard-momentum one-particle states in Weinberg

My question has essentially already been addressed in Questions concerning some parts of the section on one-particle states in Weinberg's first volume on QFT (third question), but unfortunately ...
0
votes
1answer
78 views

Quantum Mechanics: how exactly does “delta function normalization” work for eigenfunctions in 1-d free space case?

The definition of "delta function normalization" says a basis of eigenfunctions of a particle in free space are orthonormal when $$\int_{-\infty}^{\infty}\phi_n^*(\vec{r})\phi_m(\vec{r})\mathrm{d}\vec{...
1
vote
1answer
43 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 \...
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 ...
0
votes
1answer
142 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} - \...
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 ...
2
votes
2answers
96 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 ...
3
votes
1answer
103 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 ...
0
votes
0answers
26 views

Non normalisability implies uncertainty principle? [duplicate]

The wave function $\psi(x,t)$ for a free particle assuming that the position and momentum is well defined, can be solved from the schroedinger equation, $$\frac{-\hbar^2}{2m}\nabla^2\psi+V\psi=\hat{E}\...
2
votes
2answers
118 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 ...
0
votes
2answers
65 views

Show that $|n\rangle$ is correctly normalized [closed]

Prove that $$|n\rangle = \frac1{\sqrt{n!}} (\hat a^\dagger)^n |0\rangle$$ is correctly normalized. I know I must show its bra-ket equals 1 but I don't know what bra-ket notation really means, so I ...
1
vote
0answers
116 views

Problem in understanding Feynman's explanation of the Dirac-Delta function [duplicate]

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...
4
votes
1answer
89 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 ...
1
vote
2answers
422 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 $t=0$, ...
0
votes
1answer
95 views

A wave function that is normalized initially remains normalized

Suppose that $\Psi(x,t)$ is normalized at time $t=0$. Show that this implies that $\Psi(x,t)$ is normalized at all other times. I know that this makes intuitive sense, and we'd certainly want our ...
0
votes
1answer
106 views

Normalizing a wave function [closed]

A particle with mass $m$ is moving in one dimension. The wave function of the particle is $$\Psi(x,t)=Axe^{-(\sqrt{km}/2\hbar)x^2}e^{-i\sqrt{k/m}(3/2)t}$$ for $-\infty<x<\infty$, where ...
2
votes
1answer
165 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 ...
0
votes
0answers
69 views

Normalization in eq. 2.1, QFT book by T. Banks

I've recently opened "Modern Quantum Field Theory. A Concise Introduction" by Thomas Banks. One of the very first equations (about Fock space, creation/annihilation) already got me stuck wondering if ...
0
votes
0answers
83 views

QFT Normalization of multi-particle states

Peskin 7.2 states that the identity operator for the entire Hilbert space is given by $I=\ket{\Omega}\bra{\Omega}+\displaystyle\sum_\lambda\int\frac{d^3p}{(2\pi)^3}\frac{1}{2E_p(\lambda)}\ket{\...
0
votes
2answers
58 views

Boson ladder operator $n+1$ factor [closed]

Looking at Boson creation and annihilation operators, I come across that \begin{equation} b_a|n_\alpha\rangle=\sqrt{n_\alpha}|n_\alpha-1\rangle \end{equation} and \begin{equation} b_a^+|n_\alpha\...
-2
votes
1answer
87 views

How do I normalize this wavefunction? [closed]

I need to find the normalisation constant $A$ for the wave function: $$ \psi\left(x\right) = \left\{ \begin{array}{lr} A &\: \frac{-a}{4} \leq x \leq \frac{a}{4}\\ 0 &\: ...
8
votes
2answers
456 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 ...
0
votes
2answers
84 views

Normalization of the overlap $\langle x'|p'\rangle$

Let $$\langle x'|p'\rangle = N \exp(\frac{ip'x'}{\hbar})$$ be the overlap between position and momentum space, where $N$ is a normalization constant to be determined. We can then compute $N$ by $$ \...
2
votes
1answer
74 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 ...
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}}...
-3
votes
2answers
557 views

Normalizing 3-Dimensional Wave Function [closed]

How do you normalize a wave function in three dimensions with spherical coordinates?
0
votes
1answer
70 views

Normalisation of a wavefunction [closed]

If the system if found in the state: $$\psi=\sqrt{\frac{1}{2\pi}}(\frac1{\sqrt3}e^{-i3\phi}+ce^{-i4\phi})$$ what value of $c$ normalizes the wavefunction? Clearly: $$\int_0^{2\pi}\psi^*\psi d\psi=1$...
2
votes
1answer
314 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 ...
3
votes
3answers
455 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 ...
0
votes
1answer
101 views

How does one normalize this wavefunction? [closed]

Here is the question: So I could write $ N = \dfrac{1}{{\sqrt{<Ψ|Ψ>}}} $, right? Considering the parentheses in the exponential term, it looks like a good idea to switch to spherical polar ...
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: ...
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 ...
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{\...
0
votes
0answers
87 views

Simplest fermionic normalized quantum many-particle wavefunction in position representation

What is the simplest fermionic normalized quantum many-particle wavefunction, expressed in the first-quantized position representation, that you can think of? The normal single-particle examples don't ...
3
votes
3answers
994 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 ...
1
vote
1answer
953 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 &...
2
votes
1answer
267 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
2answers
256 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^*(...