Yes, the observer's wavelength is taken into account when calculating the wave function and position of an electron in quantum mechanics. In quantum mechanics, the wave function describes the probability distribution of finding a particle, such as an electron, in different locations or states.
The wave function of a particle is typically described in terms of its own reference frame or rest frame. However, when an observer is in relative motion with respect to the particle, a phenomenon known as the Doppler effect comes into play. The Doppler effect is a change in frequency or wavelength observed when there is relative motion between a source of waves (in this case, the electron) and an observer.
When an observer is moving relative to the electron, the observed wavelength and frequency of the electron's wave function will be different from those in the electron's rest frame. This is due to the relativistic effects of time dilation and length contraction.
To accurately describe the electron's wave function and position from the perspective of the moving observer, the appropriate relativistic transformations need to be applied to account for the observer's motion and the resulting changes in wavelength.
In summary, the observer's wavelength is indeed taken into account when calculating the wave function and position of an electron, particularly when there is relative motion between the observer and the electron.