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QUANTUM PHYSICS QUESTION Consider the dimensionless harmonic oscillator Hamiltonian with P (a) Show that the two wave functions yo(x) = ex²2 and 1(x) = xe-²/2 are eigenfunc- tions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the value of the coefficient a such that y2(x) = (1 + ax²)e-²/2 is orthogonal to wo(x). Then show that w2(x) is an eigenfunction of Ĥ with eigenvalue 5/2.

QUANTUM PHYSICS QUESTION Consider the dimensionless harmonic oscillator Hamiltonian with P (a) Show that the two wave functions yo(x) = ex²2 and 1(x) = xe-²/2 are eigenfunc- tions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the value of the coefficient a such that y2(x) = (1 + ax²)e-²/2 is orthogonal to wo(x). Then show that w2(x) is an eigenfunction of Ĥ with eigenvalue 5/2.

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▬▬ M 8. PC W A W 19.5= U Home Cut Copy Format Painter Clipboard File Paste | +15° |·14° |·13· |·12·1·11·1·10·|·9·1·8·1·7·1·6·1·5·1·4·1·3·1·2·1·1······20 Insert all 04:00 PM 2022-06-18 Page: 1 of 1 Words: 3 Page Layout References Calibri (Body) 11 T T Α Α΄ BIU abe X, X² A T Font I U abe X₂ X³ Document2 - Microsoft Word (Product Activation Failed) Review View B-B-S ## T AaBbCcDc AaBbCcDc AaBbC AaBbCc AaBl AaBbCcl ab Normal No Spaci... Heading 1 Heading 2 Title Subtitle F Paragraph G Styles ··2 · 1 · 1 ·|·|·|·1·1·2·1·3·1·4·1·5·1· 6 · 1 · 7 · 1 · 8 · 1 ·9·1·10° | ·11·|··12·|·13·|··14° | ・15·| ·|\_\ ·| ·17·|*18* || QUANTUM PHYSICS QUESTION Consider the dimensionless harmonic oscillator Hamiltonian  =-=-²²+½-X², with P = d dx (a) Show that the two wave functions yo(x) = e-x²/2 and y/1(x) = xe-x²/2 are eigenfunc- tions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the value of the coefficient a such that w2(x) = (1 + ax²) e-x²/2 is orthogonal to yo(x). Then show that y2(x) is an eigenfunction of Ĥ with eigenvalue 5/2. Mailings Aal Aa АА Change Styles G ■ ¥良 酒 ≡ 80% Find час ac Replace Select - Editing @? 184 V +
Transcribed Image Text:▬▬ M 8. PC W A W 19.5= U Home Cut Copy Format Painter Clipboard File Paste | +15° |·14° |·13· |·12·1·11·1·10·|·9·1·8·1·7·1·6·1·5·1·4·1·3·1·2·1·1······20 Insert all 04:00 PM 2022-06-18 Page: 1 of 1 Words: 3 Page Layout References Calibri (Body) 11 T T Α Α΄ BIU abe X, X² A T Font I U abe X₂ X³ Document2 - Microsoft Word (Product Activation Failed) Review View B-B-S ## T AaBbCcDc AaBbCcDc AaBbC AaBbCc AaBl AaBbCcl ab Normal No Spaci... Heading 1 Heading 2 Title Subtitle F Paragraph G Styles ··2 · 1 · 1 ·|·|·|·1·1·2·1·3·1·4·1·5·1· 6 · 1 · 7 · 1 · 8 · 1 ·9·1·10° | ·11·|··12·|·13·|··14° | ・15·| ·|\_\ ·| ·17·|*18* || QUANTUM PHYSICS QUESTION Consider the dimensionless harmonic oscillator Hamiltonian  =-=-²²+½-X², with P = d dx (a) Show that the two wave functions yo(x) = e-x²/2 and y/1(x) = xe-x²/2 are eigenfunc- tions of Ĥ with eigenvalues 1/2 and 3/2, respectively. (b) Find the value of the coefficient a such that w2(x) = (1 + ax²) e-x²/2 is orthogonal to yo(x). Then show that y2(x) is an eigenfunction of Ĥ with eigenvalue 5/2. Mailings Aal Aa АА Change Styles G ■ ¥良 酒 ≡ 80% Find час ac Replace Select - Editing @? 184 V +
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