By Levi A.F.J.

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**Example text**

Now 2mω turning our attention to 〈 p 2 x〉 we have 2 † † † † † h mω hmω 3h mω 〈 p 2x 〉 = ------------ 〈1 | ( bˆ – bˆ ) | 1〉 = ------------ 〈 1| – bˆ bˆ – bˆ bˆ + bˆ bˆ + bˆ bˆ | 1〉 = -------------- 2 2 2 2 h mω and one can see that for the general state |n 〉 one has 〈 pˆ x 〉 = ------------ ( 1 + 2n ) . 2 For the particular case we are interested in | n = 1〉 and the uncertainty product is 3 ∆x ∆p x = ( 〈 x 2〉 〈 p 2x 〉 ) 1 ⁄ 2 = - h 2 h For the general state | n〉 the uncertainty product ∆x ∆p x = -- ( 1 + 2 n ) 2 Solution 3 (a) We start with the reasonable assumption that the expectation value of an observable ˆ evolves smoothly in time such that associated with operator A d ˆ ∆A 〈 A〉 = ------dt ∆t which may be written as ∆A ∆t = -----------------d ˆ 〈 A〉 dt ˆ is time-independent so that In this problem the operator A d ˆ i ˆ ]〉 + 〈 ∂ Aˆ 〉 = -i- 〈 [ H ,A ˆ ]〉 〈 A〉 = -- 〈 [ H ,A h h dt ∂t since 〈 ∂ ˆ A〉 = 0 ∂t ˆ and Bˆ is In addition, the generalized uncertainty relation for operators A i ˆ , Bˆ 〉 ] ∆A ∆ B ≥ -- [ 〈 A 2 which may be re-written as ˆ , Bˆ ] 〉 2 ∆A ∆B ≥ 〈 [ A 6 If operator Bˆ is the Hamiltonian H, then ∆B corresponds to ∆ E so that A ˆ , H ]〉 = h d 〈 A ˆ〉 = h ∆ 2 ∆A ∆E ≥ 〈 [ A ------dt ∆t Since the ∆A terms on the far left and far right of the equation cancel, we have h ∆ E∆t ≥ --2 It might be tempting to make a comparison between this result and the uncertainty relation for two non-commuting operators such as momentum pˆ x and position xˆ for which h ∆ p x ∆x ≥ --- follows directly from the generalized uncertainty relation.

Both fields are needed to describe both the instantaneous state and time evolution of the Applied quantum mechanics 3 system. Quantum mechanics uses one complex wave function to describe both the instantaneous state and time evolution of the system. It is also possible to describe quantum mechanics using two coupled real wave functions corresponding to the real and imaginary parts of the complex wave function. However, such an approach is more complicated. In addition, the real and imaginary parts of the wave function have no special physical meaning.

PROBLEM 3 ˆ is time-independent but the corresponding numerical value of the Often an operator A observable A has a spread in values ∆A about an average value 〈 A ( t ) 〉 and varies with time because the system is described by a wavefunction ψ ( x , t ) which is not an eigend 〈 A ( t ) 〉 multiplied by ∆t. dt Hence, the exact time t at which the numerical value of the observable A passes through a specific value will actually have a spread in values ∆t such that state. The change in 〈 A ( t ) 〉 in time interval ∆t is the slope ∆t = ∆ A ⁄ d 〈 A〉 dt i ˆ , Bˆ ]〉 for time independent (a) Use the generalized uncertainty relation ∆A ∆B ≥ -- 〈 [ A 2 ˆ and Bˆ to show that ∆E∆ t ≥ h --- .