Energy uncertainty of quantum states

Expansion in eigenstates

Simulation

Challenges

Probability

Energy

Use the sliders to set up different quantum
states including one or more energies (with $E_{2} = 2 E_{1}, E_{3} = 3 E_{1},$ etc). Use the "?" buttons to learn more about the displayed quantities. Can you set up states with minimal/maximal energy uncertainty? Is the expectation value of energy always equal to one of the possible energy measurement outcomes? Then try the challenges in the Challenges tab!

probability

$p_{1}$ = 0.00

−

$p_{2}$ = 0.00

−

$p_{3}$ = 0.00

−

$p_{4}$ = 0.00

−

$p_{5}$ = 0.00

−

Main controls

$⟨ \hat{H} ⟩ =$ $p_{1} E_{1}$ + $p_{2} E_{2}$ + $p_{3} E_{3}$ + $p_{4} E_{4}$ + $p_{5} E_{5}$

$⟨ \hat{H} ⟩ =$ --- $E_{1}$

Energy uncertainty: $Δ E = \sqrt{⟨ {\hat{H}}^{2} ⟩ - ⟨ \hat{H} ⟩^{2}} =$ --- $E_{1}$

Show expectation value of energy
(dashed line)

Show energy uncertainty
(shaded region is ± ΔE)

Your score:

0/100

Create a superposition state (at least two energies are non-zero) for which the expectation value of energy $⟨ \hat{H} ⟩$ does not equal one of the possible measurement outcomes.