Expectation value Definition and 347 Threads

  1. facenian

    Why Does My Calculation of <x^2> Yield a Negative Result?

    Homework Statement Evaluate <x^2> for the wave function \psi(x)=\int_{-\infty}^{\infty}dk exp(-|k|/k_0)exp(ikx) My calculation yields a negative answer and I can't find my error Homework Equations...
  2. T

    Is the Expectation Value Always 1 for Normalized State Vectors?

    Homework Statement |O> = k |R1> + 1/9 |R2> a) Find k if |O> has already been normalized, and b) then the expectation value. The Attempt at a Solution a) To Normalise: |(|O>)|2 = (1/9 |R2> + k |R1>).(1/9 |R2> - k|R1>) = 1/81|R2>2 - k2|R1>2 = 1 I just assumed that |k| = (1-(1/81))0.5, but...
  3. T

    Calculating Expectation Value for z component of angular momentum

    Homework Statement Calculate the expectation value for the z component of angular momentum (operator is (h/i)(d/dx)) for the function sinx*e^(ix). Homework Equations I think the only one relevant is the expectation value: <a> = integral[psi*(a)psi] / integral[psi*psi] where psi* is...
  4. J

    I need the dirac notation expectation value explaining to me please?

    Hi, I find a lot of the time in QM i have been calculating things blindly. Take the expectation value for instance. I have worked this out in integral form plenty of times, but haven't really understood why I'm doing what I'm doing. I looked up wikipedia and apparently, for a measurable...
  5. M

    Difference between expectation value and probabilty

    Homework Statement Psi(x) = Ax -a<x<a I am trying to find the probability that my measured momentum is between h/a and 2h/a Homework Equations I have normalized A= sqrt(3/(2a^3)) I know that if I was finding the expected momentum I would use \int\Psi * p \Psi dx The...
  6. B

    Expectation value of Coulomb potential depends on relative spin

    Homework Statement Show that the expectation value of the Coulomb potential v(\vec{r_1},\vec{r_2})=\frac{e^2}{|\vec{r_1}-\vec{r_2}|}, between two electrons depends on the relative orientation of spin of the two electrons. Assume each electron is in the product state form...
  7. H

    Expectation value of momentum in discrete states

    Is there any way of proving <p> = 0 for a discrete (bound) state given it's wave function? I've seen proofs using the hermitian properties of p but I'm interested in proving that the integral of Psi*(x) Psi'(x) is identically zero regardless of Psi(x) as long as it's a solution of Schroedinger's...
  8. J

    Simple quantum problem - find eigenvalues, probabilities, expectation value?

    hi, not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) and I measure Lz, the angular momentum component...
  9. I

    Is the expectation value of this commutator zero?

    If I have H=p^2/2m+V(x), |a'> are energy eigenkets with eigenvalue E_{a'}, isn't the expectation value of [H,x] wrt |a'> not always 0? Don't I have that <a'|[H,x]|a'> = <a'|(Hx-xH)|a'> = <a'|Hx|a'> - <a'|xH|a'> = 0 ? But if I calculate the commutator, I get: <a'|[H,x]|a'> = <a'|-i p \hbar /...
  10. D

    Understanding Expectation Values in Quantum Mechanics

    Let A be an observable (opeator), and we're assuming that for a given state psi(x), the value of A is given by A acting on psi(x), namely - A|psi>. Also we assume that - P(x) = |psi(x)|^2 So, I'de expect the Expectation value of A to be defined like so: <A> = Integral[-Inf:+Inf]{ P(x) A...
  11. D

    How to Calculate the Expectation Value of H'?

    Homework Statement Calculate the expectation value of \hat{H}' in the state \psi(x,t=0). \hat{H}'=k(\hat{x}\hat{p}+\hat{p}\hat{x}) \psi(x,t=0)=A(\sqrt{3}i\varphi_{1}(x)+\varphi_{3}(x)), where A=\frac{1}{2} Homework Equations The Attempt at a Solution I know it's found by...
  12. I

    Expectation value for angular momentum

    Homework Statement A wavefunction of angular momentum states is given: \psi = \frac{1}{\sqrt{7}}|1,-1\rangle + \frac{\sqrt{35}}{7}|1,0\rangle+\sqrt{\frac{1}{7}}|1,1\rangle Calculate \langle \psi| L_{\pm} |\psi \rangle and \langle 1,1|L_+^2|\psi\rangle3. Attempt at a solution. If the...
  13. M

    Derivation of the velocity of an expectation value

    Homework Statement I am trying to derive for myself the velocity of the expectation value from the information given, specifically that <x> = \int_{-\infty}^{\infty}x|\Psi (x,t)|^2 dt (1) Eq (1) can be transformed into, \frac{d<x>}{dt} =...
  14. D

    QM Measurements - probability, expectation value

    Homework Statement What are the possible results and their probabilities for a system with l=1 in the angular momentum state u = \frac{1}{\sqrt{2}}(1 1 0)? What is the expectation value? ((1 1 0) is a vertical matrix but I can't see how to format that) Homework Equations The...
  15. C

    Expectation Value For a Given Wave Function

    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...
  16. S

    Calculating Expectation Value of Angular Momentum Squared for Hydrogen Atom

    Homework Statement Consider a hydrogen atom whose wave function at time t=0 is the following superposition of normalised energy eigenfunctions: Ψ(r,t=0)=1/3 [2ϕ100(r) -2ϕ321(r) -ϕ430(r) ] What is the expectation value of the angular momentum squared? Homework Equations I know...
  17. H

    Expectation value of spin operators.

    Homework Statement If an electron is in an eigen state with eigen vector : [1] [0] what are the expectation values of the operators S_{x}, and S_{z} Interpret answer in terms of the Stern-Gerlach experiment. The Attempt at a Solution Im not too sure how to calculate the...
  18. M

    Quantum mechanical expectation value

    I'm trying to calculate the expectation value of the momentum squared (p^2) of the harmonic oscillator ground state. The integral involves the second derivative of a Gaussian (exponential of a negative squared term) Then the integral involves, after working it out, an x^2 term times...
  19. LarryS

    Probability Density or Expectation Value?

    In a paper in Physical Review A, the author discusses a wave function for one particle, Ψ(r,t), where r is the position vector. He writes "The probability distribution for one-particle detection at a point r is given by |<r|Ψ >|2 ". Is that correct? The above expression looks, to me...
  20. L

    Calculating Expectation Value of Kinetic Energy in 3D Bound State

    ok. this is an easy enough thing to prove in one dimension but my question seems to be 3 dimensional and it's causing me some hassle: show the expectation value of the kinetic energy in a bound state described by the spherically symmetric wavefunction \psi_T(r) may be written \langle...
  21. W

    Expectation value of position of wavepacket

    Hello, this is just a general question, how is <x^2> evaluated, if <x> = triple integral of psi*(r,t).x.psi(r,t).dr (this is the expectation value of position of wavepacket) Is it possible to square a triple integral? Is <x^2> the same as <x>^2 ? I'm only wondering how the squared works...
  22. Q

    Time-dependence of expectation value <x> in a quantum harmonic oscillator?

    Find the time dependence of the expectation value <x> in a quantum harmonic oscillator, where the potential is given by V=\frac{1}{2}kx^2 I'm assuming some wavefunction of the form \Psi(x,t)=\psi(x) e^{-iEt/\hbar} When I apply the position operator, I get: <x>=\int_{-\infty}^\infty...
  23. V

    Expectation Value of Momentum Squared

    Homework Statement A particle of mass m is in the state Psi(x,t) = Ae^(-a[(mx^2)+it]) where A and a are positive real constants. a) Find A b) For what potential energy function V(x) does Psi satisfy the Shrodinger equation? c) Calculate the expectation values of x, x^2, p, and...
  24. I

    The reciprocal expectation value

    I am aware of the expectation value \left\langle\ r \right\rangle. But I was wondering what is physically meant by the expectation value: \left\langle\frac{1}{r}\right\rangle The reason I am asking is because calculating this (reciprocal) expectation value for the 1s state of hydrogen, one...
  25. Q

    Expectation value of raising/lowering operators

    Homework Statement This has been driving me CRAZY: Show that \langle a(t)\rangle = e^{-i\omega t} \langle a(0) \rangle and \langle a^{\dagger}(t)\rangle = e^{i\omega t} \langle a^{\dagger}(0) \rangle Homework Equations Raising/lowering eigenvalue equations: a |n...
  26. P

    Formula for expectation value of raidous in Hydrogen atom

    Let's consider eigenstates |nlm\rangle of hamiltonian of hydrogen atom. Can anyone prove that \langle r \rangle = \langle nlm|r|nlm\rangle = \frac{a}{2}(3 n^2-l(l+1)). Where a - bohr radious. I've been trying to prove it using some property of Laguerre polynomials (which are radial part...
  27. B

    Expectation value r^2 for a radial wave function

    Homework Statement The ground state (lowest energy) radial wave function for an electron bound to a proton to form a hydrogen atom is given by the 1s (n=1, l=0) wave function: R10 = (2 / a3/2) exp(-r / a) where r is the distance of the electron from the proton and a is a constant. a)...
  28. V

    Missing 'x' in Expectation Value Formula

    Homework Statement In my textbook, the formula for the expectation value is written as: <x> = \int \Psi^{*}\Psi dx Shouldn't there be an x next to |\Psi|^{2} ? Thanks. Homework Equations The Attempt at a Solution
  29. H

    Expectation value for a spin-half particle.

    Homework Statement Calculate the expectation value of the operator _{}Sz for a spin-half particle known to be in an eigenstate of the operator _{}Sz Homework Equations The Attempt at a Solution I know the eigenvalues for the _{}Sz but how can I find the expectation values...
  30. R

    Quantum Mechanics expectation value problem

    Homework Statement An electron is in the spin state in the Sz representation |ψ> = A (1-2i 2)T <- this is a 2 X 1 matrix If Sx is measured, what values and probabilities do you get? What is the expectation value of Sx? Homework Equations The Attempt at a Solution...
  31. J

    Angular momentum and expectation value

    My teacher said that angular momentum doesn't have orientation in space - but how can that be? Isn't cos(theta) = L_z / |L vector| ? Also (an unrelated question) could somebody give an example of how the integration process goes when you are trying to get an expectation value for something...
  32. K

    Quantum numbers of a field acquiring vacuum expectation value

    Why should symmetries require a field that acquires vacuum expectation value to have the same quantum numbers as the vacuum? Please give me a reference also..if possible...
  33. B

    Expectation value of a wave function

    Homework Statement The wave function of a state is Psi(x)= N*a(x)exp(i*p0*x/h)where a(x) is a quadratically integrable real valued function Show that the expectation value of the function is p0. Homework Equations The Attempt at a Solution The only thing I'm having a problem...
  34. A

    How do I find the variance of p for a given wave function?

    Homework Statement I am trying to find the variance of p for a wave function \Psi(x,0)=A(a^2-x^2) I'm confused about how to set up the integral. it should be something like -i^2h^2\int_{-a}^a A(a^2-x^2) (\frac{\partial\Psi}{\partial x})^2 dx I'm confused about the partial...
  35. U

    Hydrogen atom 1/r^2 expectation value

    Homework Statement Using the Feynman-Hellman theorem, determine the expectation values of 1/r and 1/r^2 for the hydrogen atom. Homework Equations Hamiltonian: H=-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}-\frac{e^2}{4\pi\epsilon_0}\frac{1}{r} energy...
  36. G

    Is this right? Re: Finding expectation value of L_z

    Okay, so I'm now reviewing ladder operators (no, not homework). While reviewing a quantum problem involving the L_z operator at this website (http://quantummechanics.ucsd.edu/ph130a/130_notes/node219.html#example:expectLz"), I found myself confused. Okay, here's my question: don't we need to...
  37. R

    Expectation Value of x: (2a+b)/4

    Homework Statement Find the expectation value <x> if: from 0 <= x <= a, psi = A x/a from a <= x <= b, psi = A(b-x)/(b-a) Normalizing gives me that A = sqrt(3/b) (verified correct)Homework Equations The Attempt at a Solution <x> = \int_0^b x \psi^2 dx = \int_0^a \frac{A^2 x^3}{a^2}dx + \int_a^b...
  38. N

    How to show that expectation value is always positive?

    Homework Statement In quantum mechanics, how to show that the expectation value is always positive? Homework Equations The Attempt at a Solution
  39. A

    Expectation value of two annihilation operators

    Hello, I was studying about the effect of a beam splitter in a text on quantum optics. I understand that if a and b represent the mode operators for the two beams incident on the splitter, then the operator for one of the outgoing beams is the following, c = \frac{(a + ib)}{\sqrt{2}}...
  40. K

    QM: expectation value of a harmonic oscillator (cont.)

    Thanks for all the help on the first question but now I have to solve for <T>. I have no idea how to do this, and I could use some help for a kick start. thanks!
  41. K

    QM: expectation value of a harmonic oscillator

    first post! but for bad reasons lol Im trying to find <x> and <p> for the nth stationary state of the harmonic potential: V(x)=(1/2)mw^2x^2 i solved for x: x=sqrt(h/2mw)((a+)+(a-)) so <x> integral of si x ((a+)+(a-)) x si. therefor the integral of si(n+1) x si + si(n-1) x si. si(n+1)...
  42. F

    What is the Expectation Value of Momentum for a Wave Function?

    Homework Statement Consider a wave function \psi (x,t) = R(x,t) exp(i S(x,t)) what is the expectation value of momentum?Homework Equations <f(x)> = \int^{\infty}_{-\infty} \psi^* f(x) \psi dx \hat{p} = -i \hbar \frac{\partial}{\partial x} The Attempt at a Solution This is for an intro to...
  43. L

    Need help with proof for expectation value relation.

    Homework Statement I have to prove the following: \hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle Edit: L = Angular Momentum & N = Torque Homework Equations I used Ehrenfest's theorem, and I've got the equation in the following form: \frac{1}{i} \left(\left[L,H\right]\right) +...
  44. K

    Finding Expectation Value of Electric Dipole Moment Matrix Form

    Homework Statement I we know the eigenstates of the system be |\psi_1\rangle and |\psi_2\rangle. Current state of the system is |\Psi\rangle = c_1 |\psi_1\rangle + c_2 |\psi_2\rangle Try to find the expectation value of electric dipole moment \mu (assume it is real) and write it in...
  45. E

    Expectation Value in Inf. Box in an Eigenstate

    Homework Statement Obtain an expression for the expectation value <Pxn>n N=1,2... of a particle in an infinite box ( V=\infty for x<0 and x>L ; V=0 for 0<X<L) which is in an eigenstate of the energy. Homework Equations Pn =+- \sqrt{2*m*En } = +- (n*pi*Hbar) / L The Attempt at a...
  46. J

    Quantum problem - Calculating the expectation value of energy?

    Homework Statement Hi all, i have a problem: i am given a time-dependent wavefunction, Ψ(x,t), and i am asked to calculate the expectation value of total energy E(t) and potential energy V(t). Ψ(x,t) = (1/sqrt2)[Ψ0(x).e-[i(E0)t/h] + Ψ1(x)e-[i(E1)t/h]], where Ψ0,1(x) are the ground and...
  47. S

    Find the expectation value of the linear momentum

    Homework Statement For a given wave function Psi(x,t)=Aexp^-(x/a)^2*exp^-iwt*sin(kx) find the expectation value of the linear momentum. Homework Equations <p>=integral(-inf,inf) psi* p^ psi dx p^=-ih(bar) d/dx sin x = (exp ix - exp -ix)/2i cos x = (exp ix + exp -ix)/2 The Attempt...
  48. K

    How Does Time Dependence Influence Expectation Values in Quantum Mechanics?

    If \Psi (x,t) = \psi (x) g(t), should I then use \Psi or \psi when calculating <p> and <p ^2>?
  49. K

    What is the Expectation Value Problem in Quantum Mechanics?

    Homework Statement Calculate \Delta x = \sqrt{\left\langle(x - \left\langle x \right\rangle )^2 \right\rangle} if \left\langle x \right\rangle = 0 and \left\langle x^2 \right\rangle = a^2(\frac{\pi - 6}{12 \pi^2}) 2. The attempt at a solution \left\langle(x - \left\langle x...
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