Change of Basis: Converting Wavefunction from S_z to S_x

In summary, the conversation is about converting a wavefunction from the S_z basis to the S_x basis. The solution is to rotate the state vector p/2 radians about the y-axis. There is some confusion about whether the new state vector should be a superposition of +x and -x.
  • #1
Norman
897
4
Hello all,

I need some help...
If I know the form of a wavefunction in the [itex] S_z [/itex] basis, say it is spin up, how do I convert that to a wavefunction expressed in the [itex] S_x [/itex] basis? Is there a very simple way to do this?
Thanks
 
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  • #2
Anyone?
Do I just rotate the spinor using a rotation operator?
Help?
 
  • #3
You've basically got it. To convert from Sz to Sx, you rotate the state vector p/2 rad about the y-axis.
 
  • #4
Tom Mattson said:
You've basically got it. To convert from Sz to Sx, you rotate the state vector p/2 rad about the y-axis.

Hi, I know this is old. I am sorry, but I have some confusion about this.

I wonder, shouldn't the new state vector be a superposition of +x and -x? If I apply the rotation operator to my z state I get a pure state.

Where has my thinking gone wrong?
 

What is a change of basis in quantum mechanics?

In quantum mechanics, a change of basis refers to the transformation of a wavefunction from one set of basis states to another set of basis states. This is often necessary when studying different physical quantities, such as angular momentum or position, as each has its own set of basis states.

Why is it important to convert a wavefunction from S_z to S_x?

Converting a wavefunction from S_z to S_x allows us to study the behavior of a system in terms of a different physical quantity. This can provide a more intuitive understanding of the system and may be necessary for certain calculations or experiments.

How is a wavefunction converted from S_z to S_x?

The conversion from S_z to S_x involves using mathematical operators, known as spin matrices, to apply a transformation to the original wavefunction. This transformation is dependent on the specific states and basis being used.

What is the relationship between S_z and S_x in quantum mechanics?

S_z and S_x are two of the three components of the angular momentum operator, with the third being S_y. These three components do not commute with each other, meaning that they cannot be simultaneously measured with complete accuracy. This is known as the Heisenberg uncertainty principle.

Are there any limitations to converting a wavefunction from S_z to S_x?

While converting a wavefunction from S_z to S_x can be a useful tool, it is important to note that it is not always possible. This is because some systems may not have well-defined states in one or both of these bases, making the conversion impossible. Additionally, the conversion can be complex and may require advanced mathematical techniques.

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