What is Wave function: Definition and 873 Discussions
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively).
The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.
For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g., z-component of spin) – these values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 1⁄2).
According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function", and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves.In Born's statistical interpretation in non-relativistic quantum mechanics,
the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle as being at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured—its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
Homework Statement
The wave function \psi_0 (x) = A e^{- \dfrac{x^2}{2L^2}}
represents the ground-state of a harmonic oscillator. (a) Show that \psi_1 (x) = L \dfrac{d}{dx} \psi_0 (x) is also a solution of Schrödinger's equation. (b) What is the energy of this new state? (c) From a look at...
Homework Statement
Find the ground state wave function for the 1-D particle in a box if V = 0 between x = -a/2 and x = a/2 and V = \infty
Homework Equations
I would guess -- Schrodinger's time-independent equation...
Homework Statement
How does the wave function of spin 1/2 change under parity?
Homework Equations
The Attempt at a Solution
The behavior of the eigenfunctions of orbital angular momentum L is easily seen from their explicit form, namely spherical function Yml is multiplied by...
N.B. I am not trying to send information back in time or generate infinite free energy (also I couldn't find how to delete the other thread but could a moderator please delete it for me)
If you could send classical information signals faster than light, then according to relativity you could...
Okay, so I am no expert in this field, but would like to acquaint myself more with quantum mechanics. As I understand it, the collapse of the wave function according to MWI only appears to occur, when really the observer is copied into as many different histories or worlds as there are possible...
Homework Statement
Calculate the probability of finding the electron in a hydrogen within the angle \pm30\circ from the x-y plane.The hydrogen is in the (2,1,1) state.
Homework Equations
probability = \int\int\int\left|R_{2,1,1}\right|^{2} \left|Y^{1}_{1}\right|^{2} r^{2} sin(\theta) dr...
I was wondering if this is correct:
\phi(k-a)=\phi(k)-\phi(a)
Where k=p/h (h bar that is) and a is some constant and \phi is the Fourier transform of a wave function (momentum function).
I know that if I had some real formula for \phi I could just test this but the problem isn't like...
Radial Wave Function; Normalized Radial Wave Function
5. Is the brute-force method the only way to show the Equation is satisfied? By that I mean differentiating R20 once, twice, and substituting it in? I tried that earlier, but it was very nasty...
Have Zernike Polynomials ever been applied to Schrondinger's Equation instead of psi? They're widely used in optics and seem to offer comparable positive traits.
http://en.wikipedia.org/wiki/Zernike_polynomials
http://wyant.optics.arizona.edu/zernikes/zernikes.htm
Lets say we have a system in a 1D infinite potential well, prepared somehow with the wavefunction: (phi)=C(a-x)x. I understand that if I try to measure the system's energy, I will collapse the system to an eigen state ((psi)=Asin(n pi x/length)+Bcos(n pi x/length)), returning an eigen energy...
Homework Statement
Here are the screenshots of the textbook...
http://i37.tinypic.com/nce7md.jpg
http://i36.tinypic.com/8zpwew.jpg
What I don't understand is the parameter (x - vt)
I am confused by the picture. This is why I can't even ask a precise question, and rather I can only...
Homework Statement
wave function of a particle in one dimension
wave function of a particle in one dimension at time t=0 given by
psi(x,0) = A(x^0.5)*(e^-ax) for x>0 or x=0
psi(x,0) = 0 for x<0
where x is in nms and a = 1nm
where is the particle most likely to be found? or for...
Imagine an infinite, positive, uniform sheet of charge with a pinhole in it. A negative particle oscillates back and forth through the pinhole and in the +-x direction. The magnitude of the force on it is constant in time (although the force reverses direction when the particle passes through...
What I know: In stationary states the time dependence is factored out so it is of the form phi(q) * e^(-i omega t), thus in its appearance there is no wave function spread. However I recall from texts that wave packet spread is considered a universal phenomena in quantum mechanics, so I am...
What are solutions to
\psi''(x) = (a_0 + a_1 x)\psi(x)
?
First idea I've had was that I could try some kind of perturbation with respect to the a_1 variable, so that
\psi(x) = A_1e^{\sqrt{a_0}x} + A_2e^{-\sqrt{a_0}x} + \psi_1(x)
would be an attempt. But I couldn't find...
I'm still wrestling with the whole uncertainty principal / wave function collapse idea. Obviously a basic building block of QM, I'm having a hard time understanding the real world evidence which supports these QM piles.
1. So from my understanding, the uncertainty principle tells us it is...
I have books (Quantum Theory by Bohm for example) with derivation of the spread of the wavefunction of a free particle in the Schrodinger equation. But does this spreading only happen as a free particle? What about under the general Schrodinger equation where there exist potentials that seem to...
I think that the wave function is the description of a particle's position at a point in time. But I'm not precisely sure what an eigenfunction is and how it is different. I know that certain eigenfunctions return certain values, and that even if you do not have a certain eigenfunction you can...
Hi. I've asked the question many times (as I'm sure many others have) why does the particle behave differently once it has been observed? Does that not mean it knows it has been observed? How does it know?
The only answer I get is: "observing destroys the wave function" , but that doesn't...
Homework Statement
Find the expectation value of x (Find <x>) given the wave function:
\psi(x)=[sqrt(m*alpha)/h_bar]e^[(-m*alpha*|x|)/(h_bar)^2]
This wave function represents the single bound state for the delta-function potential.
It's the solution to the shrodinger equation given the...
b]1. Homework Statement [/b]
I have the soloution to this question, but am confused as to what has been done between each step between lines 2,3,4. Can anyone explain how they have been simplified (espicially what happened to the operator) and what the value of the intergral is? I think I am...
It is my understanding that a measurement of S_z followed by a measurement of S_y will result in a particle which is in an eigenstate of S_y . But it appears that a measurement of say S_y followed by a measurement of S_x results in zero. I see this from a question in which I am asked to...
At time = 0 a particle is represented by the wave function
\Psi(x,0) = \left\{ \begin{array}{ccc}
A\frac{x}{a}, & if 0 \leq x \leq a, \\
A\frac{b-x}{b-a}, & if a \leq x \leq b, \\
0, & otherwise,
\end{array} \right
where A, a, and b are constants.
(a) Normalize \Psi (that is, find A, in...
Wave functions are, of course, almost always complex-valued. In all of the examples that I have seen (infinite square well, etc.), the real part of the wave function and the imaginary part of the wave function are basically the same function (except for a phase difference and possibly a sign...
Hello, I'm new to Physics Forums, so I apologize if this question seems somewhat uninformed, but I have recently started studying quantum mechanics, and was curious about the idea of the wave function collapse, and how, from what I can tell, it seems to be approached completely independently...
Homework Statement
Demonstrate that if u_1 and u_2 are solutions of the wave equation \frac{\partial ^2 u}{\partial t^2} - \triangle u=0 such that u_1 (0,x)=u_2(0,x), \partial _t u_1 (0,x)=\partial _t u_2(0,x) and such that the difference "tends to 0 at infinity" sufficiently quickly, then...
Homework Statement
Suppose that a hydrogen atom is in the 2s state. Taking r = a0, calculate value for \psi2s(a0)
Homework Equations
I did spherical harmonics for l=0 ml=0 times the radial wavefunction for n=2 l=0. Got the same thing as the solution manual attached but when I started...
Homework Statement
Consider the wave function
\Psi(x, t) = Ae^{-\lambda|x|}e^{-i\omega t}
where A, \lambda and \omega are positive real constants.
Normalize \PsiHomework Equations
\int |\Psi(x, t)|^{2} dx = 1
|\Psi(x, t)|^{2} = \Psi^{*}\PsiThe Attempt at a Solution
I have a model solution -...
Homework Statement
Using time independent 1D Shrodinger equation, show that if V(x) is even and Psi(x) is a solution, Psi(-x) is also solution. Then, assume Psi(-x) and Psi(x) differ only by a constant, show that the constant is either +1 or -1.
Homework Equations
The Attempt at a...
Homework Statement
The function
\Psi(r) = A(2-{Zr\over a})e^-{Zr\over 2a}
gives the form of the quantum mechanical wavefunction representing the electron
in a hydrogen-like atom of atomic number Z when the electron is in its first
allowed spherically symmetric excited state. Here r...
Homework Statement
http://www.ph.qmul.ac.uk/~phy319/problems/problems1.doc"
Question 2)b
The Attempt at a Solution
http://img685.imageshack.us/img685/9033/p270210111001.jpg
Is this correct?
Thanks!
Homework Statement
i. Confirming the wavefunction is normalised
ii. Calculating the expectation values: <\hat{x}> , <\hat{x^{2}}> , <\hat{p}> , <\hat{p^{2}}> as a function of \sigma
iii. Interpreting the results in regards to Heisenberg's uncertainty relation.
Homework Equations...
Hi,
I built a small program to show that the normalized hydrogen wave function (ground state) integrates to unity, as expected. But I got an absurd value: 4.6x10^19 instead.
I spanned a big volume (30 Bohr's radius) calculating and summing the product dr*dphi*dtheta * psi * psi.
Worse yet...
shai n and shai m are mutually orthogonal ...where n and m can any numerical value...but i can't imagine it how they can be perpendicular to one another ... (to me the worst thing is to think shai4 and shai100 are perpendicular) and what is the advantage or reason of it...can anyone help me to...
Homework Statement
http://img29.imageshack.us/img29/5236/phst.jpg
Homework Equations
http://img25.imageshack.us/img25/8815/11203939.jpg
I need to find the equation for the question.
The Attempt at a Solution
a)
A bunch of waves/wave functions, that have phases/amplitudes that interfere...
If I understand correctly, a measurement affects the wave function only at the moment of measurement. In other words, if one were able to compare two wave functions, one when there were measurements, and one in which there were no measurements, the only difference would be at the meager set of...
Hello
I need to calculate <x> (x is the location of neutron) and the state is
and the integral is:
Can I move the x to be near the integral symbol? and how to I multiply the matrices?
And is this true:
Can I move the psi to be near the psi*? And psi*x psi =psi^2=1 right?
thanks
Fact 1.From relativity we have come to view the universe as 4 Dimensional. That is 3 Dimensions of space and 1 of time. As such I have the following questions.
Questions
1. Given the universe is 4D, does it not follow that all objects within the universe is 4D?
2. If 1 is true, does it mean...
Homework Statement
It's not a homework problem. I'm reading my textbook (Sakurai's Modern QM), and I'm not sure about a step (eq 3.6.6 through 3.6.8). Here it is:
We start with a wave function that's been rotated:
\langle x' + y' \delta \phi, y' - x' \delta \phi, z' | \alpha \rangle
Now...
Homework Statement
A sinusoidal sound wave is described by the displacement wave function
s(x,t)=(2.00 μm)cos[(15.7 m^(-1) )x-(858 s^(-1) )t]
b) Determine the instantaneous displacement from equilibrium of the elements of air at the position x = 0.050 m at t = 3.00 ms
Homework...
Homework Statement
Hello,
Can you confirm that what I wrote is correct for the given potential?
https://www.physicsforums.com/attachment.php?attachmentid=22309&stc=1&d=1260118852
Now I wrote the term for the wave funcation and for the given symetric potential , the functions of the...
The Hydrogen Atom wave function.
With the substitution u(r) = r.R(r)
p=kr
We get a simplified version: d^2u/dp^2 = [1 - (p_0)/p + l(l + 1)/(p^2) ].u
Im sure some of you have seen that before.
Now, in the limit, p goes to infinity, I understand that we get u = A.exp[-p], but in...
I'm wondering if the wave function can be a constant in some special cases?
Now I understand that if we have a one dimensional wave function describing the location of a particle (say, along the x-axis), then the wave function can not be a constant. If it was, then it wouldn't be...
Would it be correct to say, that the observer's wave function is 'always collapsed'? I.e. that the observer can always be completely described by a bit string, while everything else only by a 'qbit string'.
Hi
According top Hunds rule I have a ^{3}P term which should be the term for the ground state for a 2p^{2} shell (in this case the outer sub-shell), this means that I have a triplet state and thus a symmetric wave function for the spinn. Since the electrons are femions the total wave function...
Hi.
Suppose that you want to fint the wave function for ionized iron which have the electron configuration 1s2 2s2 2p6 3s2 3p6 3d6
And suppose that the LS-coupling scheme gives accuret enouhgt description och the energy levels.
Paulis exclusion principle and Hunds rule gives then that the...