QM- Expectation value+Momentum.

In summary, the given wave function is used to find |ψ(x,t)|2, <x>, and <p>. However, integrating the complex conjugate of the wave function proves difficult. Any help would be appreciated.
  • #1
romsofia
595
309

Homework Statement


[tex]{\Psi (x,t)} = \frac {m \omega}{\pi h_{bar}}^{1/4}e^{- \frac {m \omega}{2h_{bar}}(x^{2}+ \frac {a^2}{2}(1+e^{-2i \omega t}+\frac {ih_{bar}t}{m}-2axe^{-i \omega t})}[/tex]

Problems: Find |ψ(x,t)|2
Compute <x> and <p>


Homework Equations


[tex]{x = \int^\infty_{-\infty} x | \Psi (x,t)|^{2} dx}[/tex]
[tex]{p = -ih_{bar} \int (\Psi^{\star} \frac {\partial \Psi}{\partial x}) dx}[/tex]



The Attempt at a Solution


I think eventually I'll be able to find the complex conjugate by just working through it. The problem really arises when I would go into integrate. I tried using Euler's rule (e^ix=isinx+cos) on the the two exponentials in the exponential in the wave function (e^-2iωt and e^-iωt).

Even after doing that, I see no way to integrate this wave-function! Any help would be appreciated on this problem.
 
Physics news on Phys.org
  • #2
|ψ(x,t)|2 = \frac {m \omega}{\pi h_{bar}}e^{- \frac {m \omega}{h_{bar}}(x^{2}+ \frac {a^2}{2}(1+cos(2 \omega t)+\frac {ih_{bar}t}{m}-2axe^{-i \omega t})}<x> = \int^\infty_{-\infty} x \frac {m \omega}{\pi h_{bar}}e^{- \frac {m \omega}{h_{bar}}(x^{2}+ \frac {a^2}{2}(1+cos(2 \omega t)+\frac {ih_{bar}t}{m}-2axe^{-i \omega t})} dx<p> = -ih_{bar} \int (\Psi^{\star} \frac {\partial \Psi}{\partial x}) dx
 

1. What is the expectation value in quantum mechanics?

The expectation value in quantum mechanics is the average value of a particular observable quantity, such as position or momentum, for a given quantum state. It is calculated by taking the integral of the observable quantity multiplied by the probability of measuring that quantity in the state.

2. How is the expectation value related to the uncertainty principle?

The uncertainty principle states that there is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known simultaneously. The expectation value provides a way to calculate the average value of a quantity, but it does not provide information about the spread or uncertainty of that quantity.

3. How is the expectation value of momentum calculated?

The expectation value of momentum is calculated by taking the integral of the momentum operator multiplied by the probability amplitude of the state. In mathematical notation, it is written as ⟨⟩, where is the bra vector and is the ket vector representing the state.

4. What is the physical significance of the expectation value?

The expectation value has physical significance because it is the average value that would be obtained if the same measurement was repeated many times on identical systems prepared in the same state. It is also the most probable measurement outcome for a given quantum state.

5. How does the expectation value change with time in quantum mechanics?

The expectation value can change with time in quantum mechanics due to the time dependence of the state vector. As the state evolves over time, the expectation value can change accordingly. This is described by the time-dependent Schrödinger equation in quantum mechanics.

Similar threads

  • Advanced Physics Homework Help
Replies
4
Views
881
  • Advanced Physics Homework Help
Replies
30
Views
1K
  • Advanced Physics Homework Help
Replies
24
Views
632
  • Advanced Physics Homework Help
Replies
10
Views
396
  • Advanced Physics Homework Help
Replies
1
Views
170
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
3
Views
829
  • Advanced Physics Homework Help
Replies
9
Views
809
  • Advanced Physics Homework Help
Replies
15
Views
1K
Back
Top