| Summary: | We consider a quantum spinless nonrelativistic charged particle moving in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mi>y</mi></mrow></semantics></math></inline-formula> plane under the action of a time-dependent magnetic field, described by means of the linear vector potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">A</mi><mo>=</mo><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mfenced separators="" open="[" close="]"><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>α</mi><mo stretchy="false">)</mo><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo></mfenced><mo>/</mo><mn>2</mn></mrow></semantics></math></inline-formula>, with two fixed values of the gauge parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> (the circular gauge) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and straight lines for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. They are expressed in terms of solutions to the classical equation of motion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover accent="true"><mi>ε</mi><mo>¨</mo></mover><mo>+</mo><msubsup><mi>ω</mi><mrow><mi>α</mi></mrow><mn>2</mn></msubsup><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mi>ε</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><msub><mi>ω</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.
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