0
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1answer
42 views

Observables in Quantum Mechanics

Studying on own quantum mechanics I came across: Preceeding text: A basic postulate of quantum mechanics tells us how to set up the operator corresponding to a given observable. Observables, ...
0
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1answer
36 views

Differences between wave function and set of orthonormal wave functions?

I'm reading a QM book. It first says for wave function: "The state of a physical system (or particle) is completely specified by an entity associated with it called a wave function, Ψ , that in ...
0
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1answer
28 views

(Level: Undergrad) Continuity Conditions on the Wavefunction and Initial Values

I know that a physically meaningful $\Psi$ needs to be continuous. However, recently I came across a problem in which they were considering a wavefunction for the infinite square well potential and ...
3
votes
2answers
104 views

What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
2
votes
3answers
104 views

I am learning Quantum Mechanics and I have some questions about some basic concept [closed]

What does a "STATE" exactly mean in quantum mechanics? What is the equivalence of "STATE" in classical mechanics? If we have a wave function $\Psi$ , its absolute square $|\Psi|^2$ is the ...
0
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2answers
109 views

Representations in quantum mechanics [closed]

This might be a very simple question. I just want someone to point me the right direction to understand things like this: $$ \langle x|x'\rangle=\delta(x-x') \\ \psi(x)=\langle x|\psi\rangle \\ ...
4
votes
4answers
247 views

Complex conjugate of momentum operator

Consider momentum operator representation in position space. $$\hat{p}=-i\frac{\partial}{\partial x} \,\ \text{and its eigen functions are } e^{ipx} \,\text{and} \,\ e^{-ipx}.$$ ...
1
vote
1answer
95 views

Wave Function Integral I need help conceptually and Mathematically

$$\int_{-\infty}^{\infty}\frac{\partial^2\bar{\psi}}{\partial{x^2}}\frac{\partial\psi}{\partial{x}}~dx.$$ I have read that this is equal to Zero. Only problem is that what I am reading about doesn't ...
1
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1answer
106 views

Singular wave function

Given a wavefunction, $\psi(x)$, is it possible for $\psi$ to be singular at a point? Are there any rules against a wavefunctions having any singularities? For instance if $$\psi(x) ...
2
votes
3answers
154 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 ...
1
vote
1answer
87 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 ...
1
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2answers
95 views

Quantum Mechanics: Momentum operator questions [closed]

I'm asked to determine $\hat{P}|\Psi_0\rangle$, $\langle{\hat{P}}\rangle$, and $\langle\hat{P}^2\rangle$ for $$\Psi_0(u) = \psi_0 + 2\psi_1$$ I understand how to make the matrix for $P$ in regards ...
4
votes
1answer
89 views

Orthogonality of summed wave functions

Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to ...
1
vote
2answers
87 views

Quantum Expectation Values

I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$: $$\langle A \rangle _\psi=\int_{\mathbb{R}}\psi^*A\hspace{0.2cm} \psi ...
1
vote
1answer
96 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 ...
3
votes
3answers
195 views

Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...
1
vote
1answer
158 views

Correspondence between wave function and state vector

I am confused with connection between state $| \psi \rangle$ of a quantum system and corresponding wave function $\psi(x)$ (at a given time). I have been told that for every state $| \psi \rangle$ we ...
4
votes
1answer
275 views

Imaginary Eigenvalue Of A Hermitian Operator

The eigenfunctions of a Hermitian operator are real. But consider a function $\psi(x)=e^{-\kappa x}$, $x\in\mathbb{R}$, where $\kappa$ is a real constant. Then, $$\hat p \psi(x)=-i\hbar ...
6
votes
2answers
439 views

Does the wave function always asymptotically approach zero?

I'm new to quantum physics (and to this site), so please bear with me. I know that quantum mechanics allows particles to appear in regions that are classically forbidden; for example, an electron ...
1
vote
2answers
240 views

General wavefunction and Schrödinger Equation

I'm starting with quantum mechanics and the book I follow (Griffiths) first introduces the wavefunction as the probability density of the position of a 0-spin single particle. Later on I've realized ...
4
votes
1answer
111 views

Can general wavefunctions be expressed as kets?

I am confused on bra-ket notation in quantum mechanics. My professor says that a ket is an eigenfunction of some operator. However, for some time now I thought a ket could represent a general ...
1
vote
1answer
453 views

Writing a Wavefunction as a Linear Combination of Eigenstates

We have the following wavefunction for the hydrogen atom: $$\psi(r,\theta,\phi)=\frac{1}{\sqrt{4\pi}}\frac{1}{(2a)^{3/2}}\frac{r}{a}e^{-r/2a}\sin(\theta)\sin(\phi)$$ where $a$ is the Bohr radius. ...
1
vote
1answer
101 views

Wavefunction's inner product

When two wavefunctions are orthogonal we can write that $$\langle\Psi_n|\Psi_m\rangle=\delta_{mn}$$ This means that $$\langle\Psi_1|\Psi_2\rangle=\langle\Psi_2|\Psi_1\rangle=0$$ But if the two ...
1
vote
2answers
313 views

Wave function decomposition

Problem: Given the wave function $\Psi_0=A\sin^2(\theta)$ along with the Hamiltonian operator of a physical system: $H=\frac{L^2}{2I}+g B L_z$, find the eigenvalues and eigenfunctions of $\hat{H}$ ...
7
votes
2answers
243 views

Quantum states as rays as opposed to vectors

I recently read that a quantum state is actually defined by a ray and not a vector. That is it is possible to multiply a state $\psi$ by any complex number $c\in \mathbb{C}$ and you won't be changing ...
2
votes
3answers
218 views

Implicit Postulate of Quantum Mechanics

Consider the following quantum system: a particle in a one dimensional box (= infinite potential well). The energy eigenstates wave functions all vanish outside the box. But the position eigenstates ...
0
votes
2answers
417 views

Basic question on bra-ket notation

Which of the following corresponds to a $ \psi(x)$, a wavefunction written in the position basis: $ x| \psi\rangle $ or $ \langle x| \psi\rangle $? If it is the second expression (which my textbook ...
4
votes
2answers
360 views

Schrödinger equation in position representation

$$ \DeclareMathOperator{\dif}{d \!} \newcommand{\ramuno}{\mathrm{i}} \newcommand{\exponent}{\mathrm{e}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle{#1}|} ...
9
votes
3answers
971 views

Meaning of inner product $\langle \vec{r} | \psi(t)\rangle $

I have come across the equation which comes out of the nothing in Zettili's book Quantum mechanics concepts and applications p. 167: $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$ ...
3
votes
2answers
772 views

Vector representation of wavefunction in quantum mechanics?

I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
1
vote
1answer
228 views

What does it mean for something to be a ket?

Ok so I will provide the following example, which I am choosing at random from Sabio et al(2010): $$\psi(r,\phi)~=~\left[ \begin{array}{c} A_1r\sin(\theta-\phi)\\ ...
2
votes
1answer
841 views

Wave function and Dirac bra-ket notation

Would anyone be able to explain the difference, technically, between wave function notation for quantum systems e.g. $\psi=\psi(x)$ and Dirac bra-ket vector notation? How do you get from one to the ...
1
vote
1answer
194 views

Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?

I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did ...
2
votes
2answers
334 views

In Dirac notation, what do the subscripts represent? (Solution for particle in a box in mind)

So the set of solutions for the particle in a box is given by $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L}).$$ In Dirac notation $<\psi_i|\psi_j>=\delta_{ij}$ assuming $|\psi_i>$ ...
0
votes
3answers
538 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. $$\int \limits^{}_{V} \left|\psi \right|^2 \, \textrm{d} V = 1\\$$ ...
1
vote
2answers
1k views

What does the quantum state of a system tell us about itself?

In quantum mechanics, quantum state refers to the state of a quantum system. A quantum state is given as a vector in a vector space, called the state vector. The state vector theoretically ...
2
votes
2answers
1k views

What's the physical significance of the inner product of two wave functions in quantum region?

I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the inner product of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the ...
4
votes
2answers
640 views

Where does the wave function of the universe live? Please describe its home

Where does the wave function of the universe live? Please describe its home. I think this is the Hilbert space of the universe. (Greater or lesser, depending on which church you belong to.) Or maybe ...