Radicalbigg 2 l sin π xl what is the probability of

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radicalbigg2Lsinπ xL.What is the probability of finding the par-ticle in the small interval Δx= 0.001Latx=L4.Correct answer: 0.001.Explanation:Let :Δx= 0.001Landx=L4.The wave function for a particle in the groundstate isφ(x) =radicalbigg2Lsinπ xL.The probability of finding the particle in theinterval ΔxisP=P(x) Δx=φ2(x) Δx=2LparenleftBigsin2π xLparenrightBigΔx .Atx=L4,P=2LparenleftBigsin2π4parenrightBig0.001L=0.001.keywords:00410.0pointsA 5.7μg particle is moving with a speed ofapproximately 0.1 cm/s in a box of length1.511 cm.Treating this as a one-dimensional particlein a box, calculate the approximate value ofthe quantum numbern.Correct answer: 2.5981×1020.Explanation:Let :h= 6.63×1034J·s,m= 5.7μg = 5.7×109kg,v= 0.1 cm/s = 0.001 m/s,andL= 1.511 cm = 0.01511 m.The energy of the particle when it is in thenth state isEn=n2h28m L2=m v22,son=2m v Lh=2 (5.7×109kg) (0.001 m/s) (0.01511 m)6.63×1034J·s=2.5981×1020.00510.0pointsIf spin is not included, how many differentwave functions correspond to the first excitedleveln= 2 for hydrogen?1.1 state2.5 states3.4 statescorrect4.2 states5.3 statesExplanation:Use the constraints onn,andmto deter-mine the number of different wave functions,excluding spin, corresponding to the first ex-cited energy state of hydrogen:Forn= 2,= 0 or 1:For= 0, m= 0 and we have 1 state;For= 1, m=-1,0,+1 and we have 3states.Hence, forn= 2 we have4 states.The four wave functions are summarized tofollowing:
hatch (heh595) – Quantum Mechanics – yeazell – (55740)3nm(n, ℓ, m)200(2,1,0)21-1(2,1,-1)210(2,1,0)211(2,1,1)
00610.0pointsThe hydrogen atom is in its ground state.What does quantum mechanics predict forthe angular momentum?(There is a differ-ence between the prediction from the quan-tum mechanic model and the Bohr model.)

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