What is Wave mechanics: Definition and 25 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.
Suppose I have a perfectly circular pool which is four meters in radius, two meters in depth, and filled with water. Say I drop a steel ball with a radius of five centimeters into the middle of the pool from a height of five meters above the water's surface. After three seconds, what will be the...
I want to simulate 2D TM scattered fields (microwave range) for austria profile. Austria profile has 2 circles beside each other of certain dielectric and one ring below the circles. So basically I have three dielectric objects in the domain of interest and also positions of Tx and Rx are known...
Why are microphones pretty good at picking up sound that is much longer in wavelength than the size of the microphone? 1khz sound has a wavelength of around a third of a meter, varying a bit depending on atmospheric conditions. Yet a 1cm diameter electret microphone can pick it up reasonably...
Homework Statement
Do waves travel faster in dense or less dense mediums?
As a wave moves from a less dense to a denser medium at a boundary end what properties change? (Wavelength, speed, frequency, amplitude...)
If waves travel faster in solids then why do we hear better through air, and if...
Is it accurate to say that interference cannot happen in Statistical Mechanics? I know it is considered a wave mechanics phenomenon but aren't waves just highly statistical ensembles, like anything else?
I always thought that Fourier says periodic spectra could be summed to create any signal...
Homework Statement
a non uniform string of length L and mass M, has a variable linear density given by μ=kx where k is the distance measured from one side of the string and k is a costant.
a) find that M=(kL2)/2
b) show that the time t required to a pulse generated from one side of the string...
In some descriptions of the Hanbury Brown and Twiss experiment, I read that the correlation results can be derived from classical wave theory, and that you only need quantum theory to explain it at the level of individual clusters of photons.
So if one knows the value of the Maxwellian field at...
Homework Statement
Let ##\langle\psi| = \int^{\infty}_{-\infty}dx\langle\psi|x\rangle\langle x|.## How do I calculate ##\langle\psi|\psi\rangle?##
Homework Equations
##\int^{\infty}_{-\infty}dxf(x)\delta(x-x_0)=f(x_0)##
The Attempt at a Solution
##\langle\psi|\psi\rangle = \int\int...
"The Heisenberg picture does not distinguish time from space, so it is better for relativistic theories than the Schrödinger equation", says Wikipedia's entry on Matrix Mechanics.
Since the Heisenberg equation of motion
\frac {dA}{dt} = \frac {i}{\hbar} [H,A] + \frac {\partial...
If this seems like a homework question please forgive me, but it is merely for understanding and confidence building. I've been reading into the Bohr Model and the Wave Mechanics Model and I read through de Broglie and have proceeded to Schrodinger, Heisenberg and Dirac. At this stage, my mind...
What wave property changes as a bullwhip wave propagates toward the tip? Wavelength or amplitude?
The problem seemed at first analogous to that of describing the behavior of a sound wave propagating through air of linearly decreasing density, except that sound is longitudinal.
Hi,
When Heisenberg proposed the Matrix Mechanics. It was totally without the concept of waves. It didn't use de Broglie idea of matter waves. In fact, Heisenberg kept fighting about the wave concept. However, Matrix Mechanics is said to be equivalent to the Schroedinger Equation that uses...
I'm in a wave mechanics class and a homework assignment asks us to describe what would happen if a driving force is applied to m1, m2, or both. The explanation should be both calculational and written. I have no idea how to model this system in equation form! Help!
Homework Statement
a) To get a wave function for a situation in which the energy is close to E_0 and the atom is almost certainly in one of the minama of the potential energy , consider the functions
\varphi_t(x)=[(\varphi_0(x)+\varphi_1(x))/(2^(1/2)))...
Homework Statement
Two operators , A and B , satisfy the equations
A=B^{\dagger}B+3 and A= BB^{\dagger}+1
a)Show that A is self adjoint
b)Find the commutator of [B^{\dagger},B]
c) Find the commutator of [B,B^{\dagger}]
d) Suppose \varphi is an eigenfunction of A with eigenvalue a...
Homework Statement
The operator Q satisfies the two equations
Q^{\dagger}Q^{\dagger}=0 , QQ^{\dagger}+Q^{\dagger}Q=1
The hamiltonian for a system is
H= \alpha*QQ^{\dagger},
Show that H is self-adjoint
b) find an expression for H^2 , the square of H , in terms of H.
c)Find the...
Homework Statement
The operators P and Q are self-adjoint and satisfy the commutation relation [P,Q]=ic where c is a real number. Show that the operator [P,Q] is anti-self-adjoint, that is , that the adjoint of the operator is the negative of the operator, consistent with the right-hand side...
Hey there. I'm trying to redo basic quantum chemistry with a lot more rigor. I'm currently using Pauling's "Introduction to Quantum Mechanics With Applications to Chemistry". Here is a copy of the page(s) I will be referring to...
In this little Dover book "Wave mechanics", by Pauli, it appears to use h for hbar, and includes a footnote right on the very first page
"1. In these lectures we use the symbol h to denote the quantity 1.05 x 10^-34 joule.sec. In the older literature this quantity was usually denoted by...
Wave Mechanics help--for optics course
Homework Statement
Consider a stretch string of length L along the x-axis in a stationary vibration mode of the form
z(x, t) = z0sin((2*pi*n/L)x) cos(omega*t)
where n is greater than or equal to 2 and is a given integer number and y0 and \omega...
Homework Statement
1. If Bohr’s theory and wave-mechanics predicts the same results for energies of hydrogen atom states, then why do we need wave mechanics with its greater complexity?
2. Compare Bohr’s theory and wave mechanics. In what respect do they differ?
3. Why don’t we observe...
Does it surprise you that the fundalmentals of wave mechanics fits so nicely into an inner product space. I assume this kind of algebra existed long ago but QM seem to fit perfectly into it. How amazing is that?
This is supposed to be an easy question, but I appear to be slightly lost. Can anyone give me a hint on what to do here?
when waves of wavelength lambda are diffracted by a circular disc of diameter D the first minimum in the intensity of the scattered waves occurs at a scattering angle z...
Hi there, i need help in a couple of questions that I'm just stumped
one of them :
A) use induction to show that
[ x (hat)^n, p(hat) sub "x" ] = i (hbar)n x(hat)^(n-1)
- so far I've figured out this equation is in relation to solve the above eq, but I'm not entirely sure how to connect the...