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Problem 7.52 Consider a spinless particle of charge q and mass m constrained to move in the xy plane under the influence of the two-dimensional harmonic oscillator potential V (x, y) = -— mw² (x² + y²). (a) Construct the ground state wave function, Vo(x, y), and write down its energy. Do the same for the (degenerate) first excited states. (b) Now imagine that we turn on a weak magnetic field of magnitude Bo pointing in the z-direction, so that (to first order in Bo) the Hamiltonian acquires an extra term H' = −μ B = . 9 2m (L.B) —— q Bo 2m (x py - y px). Treating this as a perturbation, find the first-order corrections to the energies of the ground state and first excited states.

Problem 7.52 Consider a spinless particle of charge q and mass m constrained to move in the xy plane under the influence of the two-dimensional harmonic oscillator potential V (x, y) = -— mw² (x² + y²). (a) Construct the ground state wave function, Vo(x, y), and write down its energy. Do the same for the (degenerate) first excited states. (b) Now imagine that we turn on a weak magnetic field of magnitude Bo pointing in the z-direction, so that (to first order in Bo) the Hamiltonian acquires an extra term H' = −μ B = . 9 2m (L.B) —— q Bo 2m (x py - y px). Treating this as a perturbation, find the first-order corrections to the energies of the ground state and first excited states.

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I'm solving 7.52 b) and I got Waa = 0 ( it is right answer ) but my Wab component comes out 0 too , but it is wrong . So I wonder where I made the mistake ? Wab should be ihqB0/2m

Problem 7.52 Consider a spinless particle of charge q and mass m constrained to move in the xy plane under the influence of the two-dimensional harmonic oscillator potential 1 V (x, y) = -mw² (x² + y²). 2 (a) Construct the ground state wave function, Vo(x, y), and write down its energy. Do the same for the (degenerate) first excited states. (b) Now imagine that we turn on a weak magnetic field of magnitude Bo pointing in the z-direction, so that (to first order in Bo) the Hamiltonian acquires an extra term H' = -μ B = - 9 2m (L-B) = q Bo 2m (x py - y Px). Treating this as a perturbation, find the first-order corrections to the energies of the ground state and first excited states.
Transcribed Image Text:Problem 7.52 Consider a spinless particle of charge q and mass m constrained to move in the xy plane under the influence of the two-dimensional harmonic oscillator potential 1 V (x, y) = -mw² (x² + y²). 2 (a) Construct the ground state wave function, Vo(x, y), and write down its energy. Do the same for the (degenerate) first excited states. (b) Now imagine that we turn on a weak magnetic field of magnitude Bo pointing in the z-direction, so that (to first order in Bo) the Hamiltonian acquires an extra term H' = -μ B = - 9 2m (L-B) = q Bo 2m (x py - y Px). Treating this as a perturbation, find the first-order corrections to the energies of the ground state and first excited states.
Wis = < 2° | H | 45° > Waa Wab Wba Wbb • Wag = < 1₁0 | H¹ | 1,0 > Cr 11 - a Bo zm Py Px 11> X|₁>= √= ² (α++a_) |1>= √ (9+11> +9_11>= 17 а ħ 2MW = 0 - Wig [ < 11x11 > < 0|py|0> - <1|px|l><oly10>] = √² (√₁7+ |2> + 110>) =0 V zmW = Waa= 0 -9136 zm 14 >= 11,0> 17/6 >= 10,17 Y/₁₁ - 46 (x) 4₁ (x) гр = -9 B₁ 21,01 (xpy -YPx) | 1,0 >= 2m i • Wab = < 1₂0 | H₁ | 0,17 = zmw tima (a+_a_) 11> = i 2 hmm (√2/2>-110> 2 - 91/1000 < 30 | xpy - 4px 10,17 = zm - дво 4 9 Bm [<11×10> [0lpyl + > - < 1 (p</o> <cylı >] = 2m ё биетил h X10>= √ (a₁ + a²-) 10> - √ (Voss' 113 + 0) = 2MW √ mu11>
Transcribed Image Text:Wis = < 2° | H | 45° > Waa Wab Wba Wbb • Wag = < 1₁0 | H¹ | 1,0 > Cr 11 - a Bo zm Py Px 11> X|₁>= √= ² (α++a_) |1>= √ (9+11> +9_11>= 17 а ħ 2MW = 0 - Wig [ < 11x11 > < 0|py|0> - <1|px|l><oly10>] = √² (√₁7+ |2> + 110>) =0 V zmW = Waa= 0 -9136 zm 14 >= 11,0> 17/6 >= 10,17 Y/₁₁ - 46 (x) 4₁ (x) гр = -9 B₁ 21,01 (xpy -YPx) | 1,0 >= 2m i • Wab = < 1₂0 | H₁ | 0,17 = zmw tima (a+_a_) 11> = i 2 hmm (√2/2>-110> 2 - 91/1000 < 30 | xpy - 4px 10,17 = zm - дво 4 9 Bm [<11×10> [0lpyl + > - < 1 (p</o> <cylı >] = 2m ё биетил h X10>= √ (a₁ + a²-) 10> - √ (Voss' 113 + 0) = 2MW √ mu11>
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