Unitary Operator for Transforming f to f(x-d)

[walkerj]

What Unitary operator can transfer the ground state eigenfunction
f=1/sqrt(2*pi)*exp(-x^2/2) to the ground state eigenfunction of another harmonic oscillator f=1/sqrt(2*pi)*exp(-(x-d)^2/2)?

 

[slyboy]

Your new wavefunction results from taking the original system and translating it in space. Therfore, you are looking for a translation operator. I am sure you must know which quantum observable acts as the generator of translations, so you just have to exponentiate that.

Here is a clue. Generally, spacetime symmetries lead to conserved quantitites and so translation invariance leads to the conservation of ...

 

[R0man]

The unitary operator that can transfer the ground state eigenfunction f=1/sqrt(2*pi)*exp(-x^2/2) to the ground state eigenfunction of another harmonic oscillator f=1/sqrt(2*pi)*exp(-(x-d)^2/2) is the displacement operator. This operator is defined as exp(-iPx), where P is the momentum operator and x is the displacement parameter. In this case, the displacement parameter d represents the shift in the position of the harmonic oscillator.

The displacement operator is a unitary operator, meaning it preserves the inner product of two functions. This property ensures that the transformation from f to f(x-d) will not change the normalization of the functions, making it a suitable choice for transferring the ground state eigenfunction.

Applying the displacement operator to f=1/sqrt(2*pi)*exp(-x^2/2), we get exp(-iPx)f=1/sqrt(2*pi)*exp(-x^2/2+ixd). This transformed function is equivalent to f(x-d)=1/sqrt(2*pi)*exp(-(x-d)^2/2), which is the desired ground state eigenfunction for the harmonic oscillator with a displacement of d.

In summary, the unitary displacement operator exp(-iPx) can transfer the ground state eigenfunction f=1/sqrt(2*pi)*exp(-x^2/2) to the ground state eigenfunction of another harmonic oscillator f=1/sqrt(2*pi)*exp(-(x-d)^2/2) with a displacement of d.