A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state.

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203 views

When can we assume that the wavefunction is separable

While working out the stationary states of a single particle in a 3d infinite potential box ($V=0$ inside a cuboid of known dimensions, $V=\infty$ everywhere else), I realized I had to assume the ...
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1answer
105 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 ...
0
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1answer
300 views

Calculating the expectation value of a Hamiltonian

I want to calculate the expectation value of a Hamiltonian. I have a wave function that is $$\psi = \frac{1}{\sqrt{5}}(1\phi_1 + 2\phi_2).$$ I want to know if I set this up properly. The ...
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0answers
35 views

How do I overlap this two-particle symmetric wavefunction?

Suppose we have a symmetric wavefunction that composed of a two-particle system: $$ \psi_s = \frac{1}{\sqrt 2} \left(|u,A\rangle|v,B\rangle + |v,A\rangle|u,B\rangle\right)$$ where $u_{(x)}$ and ...
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3answers
215 views

Can a wave possess spin?

Since a matter wave is associated with a particle in quantum mechanics, does the wave spins? I mean, can we visualize the spinning of wave or is it possible that the wave spins?
2
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2answers
60 views

How do I describe two entangled electrons in the same state except for a different spin

I am trying to formulate the wave function that describes two entangled electons having the same position but opposite spin. According to the Pauli exclusion principle this should be possible. And ...
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1answer
117 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 ...
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1answer
121 views

Normalizing the sum of wavefunctions and calculating probabilty - understanding concepts

A state of a particle bounded by infinite potential walls at x=0 and x=L is described by a wave function $\psi = a\phi_1 + b\phi_2 $ where $\phi_i$ are the stationary states. So let's say we want to ...
0
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3answers
250 views

Why is $|\Psi|^2$ the probability density?

I am starting with Quantum Mechanics, learning online. I can't seem to find the reason for $|\Psi|^2$ being the probability density of finding an electron. They've just taken it for granted ...
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1answer
120 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
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2answers
603 views

Calculating the most probable radius for an electron of a hydrogen atom in the ground state

This link describes a method for determining the most probable radius of an electron for a Hydrogen atom in the ground state. It states that : The radial probability density for the hydrogen ...
2
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2answers
55 views

How will a particle with energy less than $V_{\rm min}$ behave?

Consider e.g. the finite square well: $V = -V_o$ between $x=-a$ and $x=a$, $V=0$ elsewhere Now for scattering states, $E$ must be $> 0$. For normalizable bound states, $E$ must be $< 0$ and ...
2
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3answers
285 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 ...
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2answers
43 views

Potential Step - Choosing Wavefunctions

http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4190016/Quantum_Mechanics_1/ch4-virtual-book.pdf On page 2 of the above pdf they describe how they select their wavefunctions. Finding the ...
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1answer
116 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 ...
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2answers
103 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 ...
0
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1answer
60 views

Calculating the Reflection Coefficient of a Potential Step Explicitly

So I'm using the following definition for the Reflection coefficient : $$\frac{\vec{j}_{reflected}}{\vec{j}_{incident}}$$ Hence, since : $$\psi_{reflected}=Be^{-ikx}$$ and ...
0
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1answer
94 views

Solving the 1D Schrodinger Equation for a Free Particle - Two Different Methods?

So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation. ...
6
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2answers
489 views

Has the collapse of wave function due to observation been recorded?

I've seen pictures like this one, which depict the outcome of the Double-slit experiment with wave-like or particle features, depending how measurement has taken place. The graphic showing ...
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2answers
41 views

Calculating the Probability Current of a Travelling Wave

Calculate the probability current density vector $\vec{j}$ for the wave function : $$\psi = Ae^{-i(wt-kx)}.$$ From my very poor and beginner's understanding of probability density current it is : ...
2
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1answer
94 views

Electron distribution around atom when moving

I do not have much experience on this but if an atom has some electrons around nucleus and the atom itself it is moving at some speed does that affect the distribution of electrons around? I am ...
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3answers
161 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 ...
4
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1answer
113 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
votes
1answer
106 views

Apply the Heisenberg Equation to the Hamiltonian [closed]

$\frac{d}{dt}$$\hat{H}$ = $\frac{i}{\hbar}$$[\hat{H},\hat{H}]$ +$\frac{\partial{\hat{H}}}{\partial{t}}$ That's as far as I've got. I do not know much about the Heisenberg equation or even what it ...
0
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1answer
54 views

Wave function - Working towards Orthonormal solution

As you can see below, the two pages attached describe how to obtain the final solution of orthonormality. I do not completely follow the discussion but I will talk you through what I DO understand ...
4
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1answer
57 views

Effect of pressure increase on electron orbital wave functions

One of my nuclear physics exercises was to find out if increasing the pressure of a sample of $^{7}\textrm{Be}$ would increase the chance of electron capture to $^{7}\textrm{Li}$ occur. My reasoning ...
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1answer
60 views
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0answers
54 views

KdV equation and classical linear wave equation

Like we know, the standard form of KdV equation is $$u_{t}-6uu_{x}+u_{xxx}=0,\tag{1}$$ where this equation describes a solitary wave propagation and $u=u(x,t)$. On the other hand, we know the ...
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2answers
106 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 ...
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1answer
157 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 ...
2
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1answer
54 views

Free particle Schrödinger Equation

Some sources give the free-particle solution to Schrödinger equation as $$\psi(x,t) =Ae^{i(kx-\omega t)} + Be^{-i(kx+\omega t)}$$ while some sources give it as $$\psi(x,t) =Ae^{i(kx-\omega t)}$$ ...
3
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1answer
273 views

Why do we must initially assume that the wavefunction is complex?

The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial ...
8
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2answers
999 views

What is probability current in quantum mechanics?

What is probability current in quantum mechanics? Why define such a thing? I mean the meaning of probability current. I know the formula for it but I just don't get the idea of a flow of probability ...
2
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2answers
73 views

Entropy before and after collapse of the wavefunction/ and interpretation?

Seems like it might be pretty rudimentary but I want to see if my thinking is on the right track as well as what the result means. The question is, is the entropy of the collapse of a wavefunction or ...
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1answer
54 views

Reflection of an evanescent matter wave within a finite barrier?

To my understanding if I have a finite barrier with potential $V(x)>E$, then to the left of the barrier, the wavefunction can be represented as two exponentials: $$\psi= e^{(ik_{left} x)} + ...
3
votes
3answers
371 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 ...
0
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2answers
86 views

Momentum Operator in Quantum Mechanics

1) What is the difference between these two momentum operators: $\frac{\hbar}{i}\frac{\partial}{\partial x}$ and $-i\hbar\frac{\partial}{\partial x}$? How are these two operators the same? My ...
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0answers
93 views

Difference between expectation values of $L^2$, $L_z$ and measuring $L^2$, $L_z$

I was given with this hydrogen radial wavefunction $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ and was asked to find a) What are the expectation values of the ...
0
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1answer
180 views

Expectation value of energy from the position state of hydrogen atom [closed]

I was given with the position state of hydrogen atom: $$ R_{21} =\left(\sqrt{\frac{1}{3}}Y^0_1 + \sqrt{\frac{2}{3}}Y^1_1\right) $$ I am getting confused about getting the expectation value of ...
3
votes
1answer
192 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...
10
votes
2answers
268 views

The formal solution of the Schrodinger equation

Let's have Schrodinger equation (or some equation in Schrodinger form) $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ One likes to write that it has formal solution $$ \tag 2 \Psi (t) ~=~ ...
1
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1answer
281 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 ...
0
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1answer
69 views

Confusion about state of a quantum system

I am confused with the concept of state of a quantum system. First postulate of QM ussualy says that the wave function of the system contains all information about the state of the system. But reading ...
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0answers
23 views

Learning about group velocity, phase velocity and particle velocity [duplicate]

I am studying quantum physics and I would like to know a bit more in detail about group velocity, particle velocity and phase velocity. Can you guys suggest some books/online resources where I can ...
7
votes
4answers
609 views

Quantum Wave Mechanics

I am studying QM-I these days. Now, I just think of the wave function as just a mathematical function that defines the state of the particle at an instant and from it you can extract various ...
2
votes
1answer
174 views

Two ways of calculating the expectation value of momentum

The expectation value of momentum is given by: $$ \langle p\rangle = \int_{-\infty}^{\infty}\psi^{*}(x)\left(-i\hbar\frac{\partial}{\partial x}\right)\psi(x)dx $$ How can I show that the above ...
0
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1answer
69 views

Group velocity of localized wavepacket

This is a Homework problem so please feel free to not answer and just give pointers. A localized wavepacket is given as: $$\Phi(r,t=0 ) = \frac { e^{-\large\frac {r^2}{2s^2}} e^{\large\frac{i\pi ...
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1answer
66 views

Regarding derivation of Probability Current

The question for the full derivation of Probability Conservation -> Probability Current was already asked here: Probability current. I apologize for not retyping it out, but it's already beautifully ...
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1answer
40 views

Conduction and propagation [duplicate]

What is the difference between conduction of electric wave in conductor and propagation of electromagnetic wave in dielectric? Why propagation term is used for dielectric and conduction for ...
0
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2answers
373 views

Schrödinger's Equation and the depth of a finite potential well

Before I ask my question, I have to stress: I have absolutely no idea what the math is going on. I've read my textbook, several Wikipedia articles, scoured the internet, and don't feel anymore ...