Eigenvalues dependent on choice of $\vec{A}$?

In summary, the two energies found for the magnetic field given by: $$\vec{A}=B(-y,0,0)$$ and $$\vec{A}=\frac{B}{2}(-y,x,0)$$ are both the same, since the wavefunction "absorbs" the small difference in the magnetic field.
  • #1
CharlieCW
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5

Homework Statement



A particle with spin s=1/2 moves under the influence of a magnetic field given by:

$$\vec{A}=B(-y,0,0)$$

Find the eigenvalues of the corresponding Pauli hamiltonian. Repeat the same process for:

$$\vec{A}=\frac{B}{2}(-y,x,0)$$

Explain your result by relating the wavefunctions of both cases.

Homework Equations



$$\vec{\nabla}\times \vec{A}$$

$$\hat{H}=H_0+\frac{1}{2m}(-e\vec{A}\cdot\vec{p}+i\hbar\vec{\nabla}\cdot\vec{A}+e^2\vec{A}^2)+e\phi+\mu_b \vec{\sigma}\cdot\vec{B}$$

$$L_z=-ih(x\partial_y-y\partial_x)$$

$$\mu_B=\frac{e\hbar}{2m}$$

$$\hat{H}\Psi(n,l,m)=E\Psi(n,l,m)$$

$$L_z\Psi(n,l,m)=m\hbar\Psi(n,l,m)$$

The Attempt at a Solution



My question is actually punctual as I've solved most of the exercise. My problem arises in the hamiltonian used to calculate the energies, as they should be the same but the calculations say otherwise.

We can easily verify that, for both cases, we have the same magnetic field ##\vec{B}=B\hat{z}##, which is not surprising since the vector potential is not unique for a given field.

Let's begin with the second case with ##\vec{A}=\frac{B}{2}(-y,x,0)## since it's easier. Calculating some important products:

$$\vec{A}\cdot\vec{p}=A_xp_x+A_yp_y=-Bi\hbar(y\partial_x-x\partial_y)=B(L_z)$$

$$\vec{\nabla}\cdot\vec{A}=0$$

Substituting in the Pauli hamiltonian (with ##\phi=0## and ##e\rightarrow-e## for an electron, and neglecting second order terms in ##\vec{A}##):

$$\hat{H}=H_0+\frac{1}{2m}(eB(L_z))+e\phi+\mu_b \vec{\sigma}\cdot\vec{B}$$

$$\hat{H}=H_0+\frac{\mu_B B}{\hbar}(L_z+\sigma_z)$$

Substituting our hamiltonian in the time-independent Schrodinger equation, we find straightforward the energies by applying the operators to a hydrogen-like wavefunction:

$$(H_0+\frac{\mu_B B}{\hbar}(L_z+\sigma_z))\Psi(n,l,m)=E\Psi(n,l,m)$$

Since the Pauli matrix ##\sigma_z=(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array})##, we get 2 equations with only a sign difference on the last term, so we get two pairs of energies:

$$E^{\pm}_{nm}=E_n+\mu_B B (m\pm 1)$$

Now for the first case, with ##\vec{A}=\frac{B}{2}(-y,0,0)##, the procedure is similar except for one key difference. Let's calculate the relevant products again:

$$\vec{A}\cdot\vec{p}=A_xp_x=B(i\hbar y\partial_x)=B(L_z+i\hbar x\partial_y)$$

$$\vec{\nabla}\cdot\vec{A}=0$$

Now the hamiltonian has the form:

$$\hat{H}=H_0+\frac{1}{2m}(eB(L_z+i\hbar x\partial_y))+e\phi+\mu_b \vec{\sigma}\cdot\vec{B}$$

Substituting into time-independent Schrodinger, we finally get:

$$H_0\Psi'(n,l,m)+\frac{\mu_B B}{\hbar}(L_z+\sigma_z+i\hbar x\partial_y)\Psi'(n,l,m)=E\Psi'(n,l,m)$$

Which seems to don't give the same eigenvalues (I'm not sure how to evaluate the last term ##(i\hbar x\partial_y)\Psi(r)##). However, this doesn't make sense, since the energies shouldn't depend on the choice of the vector potential but on the external magnetic field, which is the same for both cases.

My idea is that the wavefunction should "absorb" this small difference so both energies (eigenvalues) are the same, but I'm not sure how to proceed here. Or if there's a way to act the last term on the eigenfunction.
 
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  • #2
Solved it.
 

Related to Eigenvalues dependent on choice of $\vec{A}$?

1. What are eigenvalues dependent on?

Eigenvalues are dependent on a matrix, specifically the choice of matrix A. They are a property of a matrix and can vary depending on the values in the matrix.

2. How do eigenvalues change with different choices of matrix A?

Eigenvalues can change with different choices of matrix A because they are a property of the matrix. If the values in the matrix are changed, the eigenvalues will also change.

3. Can the eigenvalues of a matrix A be the same for different choices of A?

Yes, it is possible for the eigenvalues of different matrices to be the same. This can happen if the matrices have similar structures or if they share common characteristics.

4. Are eigenvalues dependent on the size of the matrix A?

Yes, eigenvalues are dependent on the size of the matrix A. If the size of the matrix is changed, the eigenvalues will also change. A larger matrix may have more eigenvalues than a smaller matrix.

5. How can understanding eigenvalues dependent on choice of matrix A be useful in science?

Understanding eigenvalues dependent on choice of matrix A can be useful in many areas of science, such as physics, engineering, and data analysis. It can help in solving systems of equations, understanding the stability of dynamic systems, and finding patterns in data. It is also a fundamental concept in linear algebra and is used in many applications of mathematics.

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